REFINEMENTS, GENERALIZATIONS, AND

REFINEMENTS, GENERALIZATIONS, AND APPLICATIONS OF JORDAN’S INEQUALITY AND RELATED PROBLEMS FENG QI, DA-WEI NIU, AND BAI-NI GUO

Abstract. This is a survey and expository article. Some new developments on refinements, generalizations, and applications of Jordan’s inequality and related problems, including some results about Wilker-Anglesio’s inequality, some estimates for three kinds of complete elliptic integrals, and several inequalities for the remainder of power series expansion of the exponential function, are summarized. This article is a slightly revised and updated version of the preprint [96] and the survey article [116].

Contents 1. Jordan’s inequality and related inequalities 1.1. Jordan’s inequality 1.2. Kober’s inequality 1.3. Redheffer-Williams’s inequality and Li-Li’s refinement 1.4. Mercer-Caccia’s inequality 1.5. Prestin’s inequality 1.6. Some inequalities obtained from Taylor’s formula 1.7. Cusa-Huygens’s and related inequalities 1.8. Some inequalities related to trigonometric functions 2. Refinements and generalizations of Jordan’s and related inequalities 2.1. Qi-Guo’s refinements of Kober’s and Jordan’s inequality 2.2. Refinements of Jordan’s inequality by L’Hˆospital’s rule 2.3. Li’s power series expansion and refinements of Jordan’s inequality 2.4. Li-Li’s refinements and generalizations 2.5. Some generalizations of Redheffer-Williams’s inequality 2.6. Some generalizations and related results 3. Refinements of Jordan’s inequality and Yang’s inequality 3.1. Yang’s inequality 3.2. Zhao’s result 3.3. Debnath-Zhao’s result ¨ 3.4. Ozban’s result 3.5. Jiang-Hua’s results 3.6. Agarwal-Kim-Sen’s result 3.7. Zhu’s results

2 2 3 3 4 4 4 5 5 6 6 8 9 9 11 12 13 13 13 14 14 14 15 15

2000 Mathematics Subject Classification. 26D05, 26D15, 33B10, 33B15, 33C10, 33C15, 33C75. Key words and phrases. Jordan’s inequality, Kober’s inequality, Yang’s inequality, refinement, generalization, application, L’Hˆ ospital’s rule, auxiliary function, Wilker-Anglesio’s inequality, sine function, cosine function, Bessel function, complete elliptic integral, hyperbolically trigonometric function, exponential function. 1

2

F. QI, D.-W. NIU, AND B.-N. GUO

3.8. Niu-Huo-Cao-Qi’s result 4. Generalizations of Jordan’s inequality and applications 4.1. Qi-Niu-Cao’s generalization and application 4.2. Zhu’s generalizations and applications 4.3. Wu’s generalization and applications 4.4. Wu-Debnath’s generalizations and applications 4.5. Wu-Srivastava’s generalizations and applications 4.6. Wu-Debnath’s general generalizations and applications 4.7. Wu-Srivastava-Debnath’s generalization and applications 5. Refinements of Kober’s inequality 5.1. Niu’s results 5.2. Zhu’s result 6. Niu’s applications and analysis of coefficients 6.1. An application to the gamma function 6.2. Applications to definite integrals 6.3. Analysis of coefficients 6.4. A power series 6.5. A remark 7. Generalizations of Jordan’s inequality to Bessel functions 7.1. Neuman’s generalizations of Jordan’s inequality 7.2. Niu-Huo-Cao-Qi’s generalizations of Jordan’s inequality 7.3. Baricz’s generalizations of Cusa-Huygens’s inequality 7.4. Baricz’s generalizations of Redheffer-Williams’s inequality 7.5. Lazarevi´c’s inequality and generalizations 7.6. Oppenheim’s problem 7.7. Some inequalities of Bessel functions 8. Wilker-Anglesio’s inequality and its generalizations 8.1. Wilker’s inequality and generalizations 8.2. Wilker-Anglesio’s inequality 8.3. An open problem 9. Applications of a method of auxiliary functions 9.1. Estimates for a concrete complete elliptic integral 9.2. Estimates for the second kind of complete elliptic integrals 9.3. Inequalities for the remainder of power series expansion of ex 10. Estimates and inequalities for complete elliptic integrals 10.1. Inequalities between three kinds of complete elliptic integrals 10.2. Carlson-Vuorinen’s inequality 10.3. Some recent results of elliptic integrals Acknowledgements References

16 17 17 18 20 20 23 24 24 25 25 25 25 25 26 26 26 26 27 27 27 28 28 28 29 30 30 30 31 31 31 32 33 34 34 34 35 35 36 36

1. Jordan’s inequality and related inequalities 1.1. Jordan’s inequality. The well-known Jordan’s inequality (see [23, p. 143], [36], [67, p. 269] and [79, p. 33]) reads that sin x 2 ≤ 0, ∞ k sin x 2 X Rk (π/2) 2 = + π − 4x2 , (2.18) (−1)k x π k!π 2k k=1

where Rk (x) =

∞ X n=k

satisfy (−1)k Rk

π 2

(−1)n n! x2n (2n + 1)!(n − k)!

(2.19)

> 0 and 0 x x sin x and Rk+1 (x) = −kRk (x) + Rk0 (x) (2.20) R1 (x) = 2 x 2 for k ∈ N. As a direct consequence of the above identity, the following lower bound for the function sinx x was established in [70]: 12 − π 2 2 3 2 1 sin x 10 − π 2 2 2 2 ≥ + 3 π 2 − 4x2 + π − 4x π − 4x2 + 5 7 x π π 16π 16π 4 π π 4 − 180π 2 + 1680 2 π − 4x2 , 0 < x < . (2.21) + 9 3072π 2 Equality in (2.21) is valid if and only if x = π2 . The constants 4 2 +1680 and π −180π are the best possible. 3072π 9 Moreover, by employing ∞ X x 22k − 2 2k = (−1)k+1 B2k x sin x (2k)!

1 12−π 2 10−π 2 π 3 , 16π 5 , 16π 7

(2.22)

k=0

for |x| < π, where B2k for 0 ≤ k < ∞ is the well-known Bernoulli’s numbers, it was presented in [70] that x 1 7 4 31 6 ≤ 1 + x2 + x + x , |x| < π. (2.23) sin x 6 360 15120 2.4. Li-Li’s refinements and generalizations. In [71], two seemingly general but not much significant results for refining or generalizing Jordan’s inequality (1.1) were discovered. 2.4.1. The first result may be stated as follows: If the function g : 0, π2 → [0, 1] is continuous and sin x ≥ g(x) (2.24) x π for x ∈ 0, 2 , then the double inequality 2 π sin x −h + h(x) ≤ ≤ 1 + h(x) (2.25) π 2 x for x ∈ 0, π2 holds with equality if and only if x = π2 , where Z x Z u h πi 1 2 v g(v) d v d u, x ∈ 0, . (2.26) h(x) = − 2 2 0 0 u

10


Since g(x) is positive, it is clear that the function h(x) is decreasing and negative, therefore, the double inequality (2.25) refines Jordan’s inequality (1.1). Remark 2.9. It is remarked that the upper bound in (2.25) was not considered in [71], although it is implied in the arguments. On the other hand, if the inequality (2.24) is reversed, then so is the inequality (2.25). Remark 2.10. Upon taking g(x) = 0 in (2.24) and (2.26), Jordan’s inequality (1.1) is derived from (2.25). If letting g(x) = π2 , then inequalities (1.7) and πi sin x 1 2 (2.27) ≤1− x , x ∈ 0, x 3π 2 are deduced from (2.25). If choosing g(x) as the function in the right-hand side of (1.7), then the inequality πi 2 2 60 + π 2 2 1 sin x ≥ + π − 4x2 + π 2 − 4x2 , x ∈ 0, (2.28) x π 720π 960π 2 follows from the left-hand side of (2.25). These three examples given in [71] seemly show that, by using some lower bound for sinx x on 0, π2 , a corresponding stronger lower bound may be derived from the left-hand side inequality in (2.25). Actually, this is not always valid: By taking g(x) as the function in the right-hand side of (1.8) or the one in the left-hand side of (2.12), it was obtained that πi 2 sin x 2 1 1 ≥ + π 2 − 4x2 + π 2 − 4x2 , x ∈ 0, . (2.29) 3 x π 60π 80π 2 Unluckily, the inequality (2.29) is worse than both the inequality (1.8) and the left-hand side inequality in (2.12). This tells us that the inequality πi π 2 −h (2.30) + h(x) > g(x), x ∈ 0, π 2 2 is not always sound. Therefore, Theorem 2.1 in [71], one of the main results in [71], is not always significant and meaningful. This reminds us of proposing a question: Under what conditions on 0 < g(x) < 1 for x ∈ 0, π2 the inequality (2.30) holds? 2.4.2. The second result in [71] is procured basing on Lemma 2.1. It can be summarized as follows: If the function f (x) ∈ C 2 0, π2 satisfies f 0 (x) > 0 and 2 0 0 x f (x) 6= 0 for x ∈ 0, π2 , then the double inequality lim+

x→0

π i sin x sin x/x − 2/π h 2 f (x) − f ≤ − f (x) − f (π/2) 2 x π π i sin x/x − 2/π h ≤ lim f (x) − f , 2 x→(π/2)− f (x) − f (π/2)

πi x ∈ 0, 2

(2.31)

is sharp in the sense that the limits before brackets in (2.31) cannot be replaced 0 by larger or smaller numbers. If f 0 (x) < 0 and x2 f 0 (x) 6= 0, then the inequality (2.31) is reversed. As an application, by taking f (x) = xn for n ∈ N in (2.31), the inequality (2.8) and sin x 2 2 n 2 π − 2 + π − (2x)n ≤ ≤ + n+1 π n − (2x)n , n ≥ 2 (2.32) n+1 π nπ x π π were showed in [71, Theorem 3.2], where the equalities hold if and only if x = π2 and the constants nπ2n+1 and ππ−2 n+1 in (2.32) are the best possible.

JORDAN’S INEQUALITY AND RELATED PROBLEMS

11

If taking n = 2 in (2.32), then the inequalities (1.8) and (2.6) are recovered. The inequality (2.32) for n = 3 and n = 4 respectively implies (2.11) and (3.11). Remark 2.11. What essentially established in Section 3 of [71] are sufficient condi π x/x−2/π tions for the function fsin (x)−f (π/2) to be monotonic on 0, 2 . Generally, more new sufficient conditions may be further found. 2.5. Some generalizations of Redheffer-Williams’s inequality. 2.5.1. Chen-Zhao-Qi’s results. In [31, 32], the following three inequalities similar to (1.5) were established: If |x| ≤ 12 , then cos(πx) ≥

1 − 4x2 1 + 4x2

and

cosh(πx) ≤

1 + 4x2 ; 1 − 4x2

(2.33)

If 0 < |x| < 1, then sinh(πx) 1 + x2 ≤ . πx 1 − x2

(2.34)

2.5.2. Zhu-Sun’s results. In [170], by using Lemma 2.1 and other techniques, the above three inequalities are sharpened and some new results were demonstrated as follow: (1) The double inequality 2 2 β α π − x2 π − x2 sin x ≤ ≤ π 2 + x2 x π 2 + x2

(2.35)

2

holds for 0 < x < π if and only if α ≤ π12 and β ≥ 1. (2) The double inequality 2 β 2 α π − 4x2 π − 4x2 ≤ cos x ≤ π 2 + 4x2 π 2 + 4x2

(2.36)

2

holds for 0 ≤ x ≤ π2 if and only if α ≤ π16 and β ≥ 1. (3) The double inequality 2 α 2 β π + 4x2 tan x π + 4x2 ≤ ≤ π 2 − 4x2 x π 2 − 4x2

(2.37)

2

holds for 0 < x < π2 if and only if α ≤ π24 and β ≥ 1. (4) The double inequality 2 α 2 β r + x2 sinh x r + x2 ≤ ≤ r 2 − x2 x r 2 − x2

(2.38)

2

r . holds for 0 < x < r if and only if α ≤ 0 and β ≥ 12 (5) The double inequality 2 α 2 β r + x2 r + x2 ≤ cosh x ≤ r 2 − x2 r 2 − x2

holds for 0 ≤ x < r if and only if α ≤ 0 and β ≥

r2 4 .

(2.39)

12


(6) The double inequality 2 β α 2 r − x2 tanh x r − x2 ≤ ≤ r 2 + x2 x r 2 + x2 holds for 0 < x < r if and only if α ≤ 0 and β ≥

(2.40)

r2 6 .

2.5.3. Zhu’s sharp inequalities. In [168], using Lemma 2.1 and other techniques, some sharp inequalities for the sine, cosine and tangent functions are presented as follows: (1) For x ∈ (0, π], the double inequality 2 α β 2 π − x2 sin x π − x2 √ ≤ (2.41) ≤ √ x π 4 + 3x4 π 4 + 3x4 2

holds if and only if α ≥ π6 and β ≤ 1. (2) The following inequalities are valid: 2 π2 /6 2 3/4 π − 4x2 π − 4x2 π √ ≤ cos x ≤ √ , 01> > > csc α|x| β|x| β|x| α|x| α|x| 2α

(2.48)

for 0 < β < α and 0 < |αx| < π2 . In [124], it was proved that a positive and concave function is logarithmically concave and that the function sinx x for 0 < x < π2 is a concave function. As a corollary, the following inequality was obtained: 2 2(2 − π) π sin x x≥ , 0 0 and B > 0 with A + B ≤ π, where the equality holds if and only if λ = 0 or A + B = π. Remark 3.1. The inequality (3.1) has been generalized in [154, 156] and related references therein. 3.2. Zhao’s result. In [155, Theorem 1 and Theorem 2], by using inequalities (1.1) and (1.5) respectively, it was concluded that X n 2 π n 2 2 4 λ cos2 λ ≤ Hij ≤ π λ (3.2) 2 2 2 1≤i 0 and some known inequalities were derived. x 4.7. Wu-Srivastava-Debnath’s generalization and applications. In virtue of Lemma 2.1, the following conclusion for bounding the function sinx x was gained √ in [145]: For n ∈ N, 0 < x ≤ θ ≤ π and f (x) = sin√x x , we have n−1 n k f (n) (θ2 ) 2 sin x X f (k) (θ2 ) 2 x − θ2 ≤ − x − θ2 n! x k! k=0 # " n−1 X (−1)k θ2k f (k) (θ2 ) n 1 ≤ 2n 1 − θ2 − x2 . (4.55) θ k! k=0

The equalities in (4.55) hold true if and only if x√= θ. In [145, Lemma 3], the function f (x) = sin√x x was proved to be completely monotonic on (0, π 2 ]. For detailed information on the class of completely monotonic functions, please see the survey papers [93, 94, 112, 113] and related references therein. In the final of [145], Yang’s inequality (3.1) was generalized by virtute of the inequality (4.55) for n = 4 and θ = π2 .


25

5. Refinements of Kober’s inequality 5.1. Niu’s results. As a direct consequence of (3.28), the following general refinements of Kober’s inequality was obtained in [84]: For 0 < x ≤ π2 , k ∈ N and n ∈ N, inequalities " # n π 2 X k k −x + αk (4x) (π − x) ≤ cos x 2 π k=1 " # n 2 X π k k −x + βk (4x) (π − x) , (5.1) ≤ 2 π k=1

which may be deduced by replacing x with x − π2 in (3.28), and n X k X (−4)i ki αk π 2k−2i 2i+2 x2 x ≤ 1 − cos x − 2i + 2 π k=1 i=0 n X k X (−4)i ki βk π 2k−2i 2i+2 ≤ x , (5.2) 2i + 2 k=1 i=0 which follows from integrating (3.28) from 0 to x ∈ 0, π2 , hold with constants αk and βk defined by (3.29) and (3.30) respectively. 5.2. Zhu’s result. By an utilization of the inequality (3.21) and a simple transformation of variables, the following Kober type inequality was deduced in [159, Theorem 13]: Let n X ak 2k+1 π (1 − u)uk (2 − u)k (5.3) R(u) = 2 k=0

and

1 2n+3 π (1 − u)un+1 (2 − u)n+1 2 for n ≥ 0, where ak for k ≥ 0 are defined by (3.23). Then the inequality πu R(u) + λS(u) ≤ cos ≤ R(u) + µS(u) 2 S(u) =

holds if either 0 ≤ u ≤ 1, λ = an+1 and µ = λ=

1−

Pn

k=0 ak π π 2(n+1)

2k

P 2k 1− n k=0 ak π π 2(n+1)

(5.4)

(5.5)

or 1 ≤ u ≤ 2 and

and µ = an+1 .

6. Niu’s applications and analysis of coefficients 6.1. An application to the gamma function. In [84], combining πz Γ(1 + z)Γ(1 − z) = sin πz with (3.28) yields that if 0 < x < π2 and n ∈ N then n

n

k=1

k=1

(6.1)

k k 2 X 1 2 X + αk π 2 − 4x2 ≤ ≤ + βk π 2 − 4x2 , (6.2) π Γ(1 + x/π)Γ(1 − x/π) π where Γ(x) is the classical Euler gamma function defined for x > 0 by Z ∞ Γ(x) = e−t tx−1 d t. 0

(6.3)

26


6.2. Applications to definite integrals. In [84], as applications of (3.28), the following conclusions were also obtained: (1) For 0 < x ≤

π 2

and k, n ∈ N,

Z x n X k X (−4)i ki αk π 2k−2i 2i+1 sin t 2 x+ x ≤ dt π 2i + 1 t 0 i=0 k=1

n X k X (−4)i ki βk π 2k−2i 2i+1 2 ≤ x+ x . (6.4) π 2i + 1 i=0 k=1

(2) Let f (x) be continuous on [a, b] such that f (x) 6≡ 0 and 0 ≤ f (x) ≤ M . If 0 < b − a < π and n ∈ N, then Z 0
1, √ π 1 < αk < 2k √ , (6.6) − 2k √ π 4k + 1 π 4k + 1 √ √ 1 − 2/π + π k − 1 − 1/2 βk < , (6.7) π 2k √ √ 1 − 2/π + π k − 1/2 0 ≤ βk − αk < . (6.8) π 2k Recently, some more accurate estimates of the coefficients αk and βk are carried out in [85]. 6.4. A power series. The inequality (3.28) can be rearranged as n

n

k=1

k=1

k X k sin x 2 X 0≤ − − αk π 2 − 4x2 ≤ (βk − αk ) π 2 − 4x2 → 0 x π as n → ∞, this implies that ∞

sin x =

X k 2 x− αk x π 2 − 4x2 . π

(6.9)

k=1

This gives an alternative power series expansion similar to (2.18) and (3.26). 6.5. A remark. It is natural to consider that the series (2.18), (3.26) and (6.9) should be the same one, although they seems to have different expressions. This was affirmed in [85], among other things.


27

7. Generalizations of Jordan’s inequality to Bessel functions For x ∈ R, some Bessel functions are defined by ∞ x 2n+p X (−1)n Jp (x) = , n!Γ(p + n + 1) 2 n=0 ∞ X

x 2n+p 1 , n!Γ(p + n + 1) 2 n=0 ∞ X (−1)n cn Γ p + b+1 x 2n 2 λp (x) = , b+1 2 n!Γ p + 2 + n n=0 Ip (x) =

Jp (x) = 2p Γ(p + 1)x−p Jp (x),

(7.1) (7.2) (7.3) (7.4)

Ip (x) = 2 Γ(p + 1)x

−p

J−1/2 (x) = cos x,

I−1/2 (x) = cosh x,

(7.6)

sin x , J1/2 (x) = x

sinh x I1/2 (x) = . x

(7.7)

p

Ip (x).

(7.5)

It is well-known that

7.1. Neuman’s generalizations of Jordan’s inequality. In [82], it was established for p ≥ 12 and |x| ≤ π2 that s " !# ! 3 x 1 2p+1+(p+2) cos x ≥ Jp (x) ≥ cos p . (7.8) 3(p + 1) 2(p + 2) 2(p + 1) When p = − 12 equality in (7.8) validates. Taking in (7.8) p = 12 leads to " r !# 2 5 3 sin x x 2 + cos x ≥ ≥ cos √ , 9 2 5 x 3

h π πi x∈ − , . 2 2

(7.9)

By employing Lemma 2.1, inequalities (2.2) and (2.12) are generalized in [12] as π h cπ π i π − 2x h π i π − 2x ≤ λp (x) − λp ≤ λp+1 (7.10) 1 − λp 2 π 2 2k 2 π for k ≥ 12 and 0 ≤ c ≤ 1 and h c π i π 2 − 4x2 π h π i π 2 − 4x2 (7.11) λp+1 ≤ λp (x) − λp ≤ 1 − λp 4k 2 4 2 2 π2 for k ≥ 0 and 0 ≤ c ≤ 1. In [10], inequalities (7.10) and (7.11) were further improved. 7.2. Niu-Huo-Cao-Qi’s generalizations of Jordan’s inequality. In [84, 85], the following two conclusions were established: (1) For n ∈ N and x ∈ (0, π2 ], if k ≥ 12 and 0 ≤ c ≤ 1, then n X

γi π 2 − 4x2

i=0

where γi =

i

≤ λp (x) ≤

n X

i ηi π 2 − 4x2 ,

(7.12)

i=0

c i Γ(k) λi+p π/2 , 16 i!Γ(k + i)

0 ≤ i ≤ n,

(7.13)

28


and  γ i , 0 ≤ i ≤ n − 1, ηi = 1 − Pn−1 γ` π 2` (7.14) `=0  , i = n, 2n π are the best possible. For k > 0, c ≤ 0 and x ∈ 0, π2 , when n is odd the inequality (7.12) holds, when n is even the inequality (7.12) is reversed. (2) For n ∈ N and 0 < x ≤ θ ≤ π2 , if k ≥ 12 and 0 ≤ c ≤ 1, then n X

σi θ 2 − x2

i=0

i

≤ λp (x) ≤

n X

i νi θ2 − x2 ,

(7.15)

0≤i≤n

(7.16)

i=0

where σi =

c i 4

Γ(k) λi+p (θ), i!Γ(k + i)

and  σi , 0 ≤ i ≤ n − 1, νi = 1 − Pn−1 σ` θ2` (7.17) `=0  , i=n 2n θ are the best possible. For k > 0, c ≤ 0 and 0 < x ≤ θ < ∞, if n is odd the inequality (7.15) holds true, if n is even the inequality (7.15) is reversed. We remark that for c ∈ [0, 1] the conditions on x and k can be relaxed, as it was stated in [19, pp. 123–124]. 7.3. Baricz’s generalizations of Cusa-Huygens’s inequality. The inequality (1.18) was generalized in [12] to 1 + 2akλp (x) 1 + 2akλp (x) ≤ λp+1 (x) ≤ , (7.18) a(2k − 1) + π/2 a + 1 + a(2k − 1) √ where |x c | ≤ π2 , a ∈ (0, 12 , c ≥ 0, and k ≥ 21 . By making use of the inequality (1.16) and (1.17), the inequality (7.18) was further strengthened as 1 + 2kλp (x) 1 + kλp (x) ≤ λp+1 (x) ≤ . 2k + 1 k+1

(7.19)

7.4. Baricz’s generalizations of Redheffer-Williams’s inequality. In [11], inequalities (1.5), (2.33) and (2.34) were generalized to the case of Bessel functions. The motivation of the paper [11] comes from [31, 32, 96, 97, 109] and other related references. 7.5. Lazarevi´ c’s inequality and generalizations. An inequality due to [68] states that 3 sinh t > cosh t (7.20) t for t 6= 0. The exponent 3 in (7.20) is the best possible. See also [23, p. 131], [67, p. 300] and [79, p. 270]. In [4, pp. 808–809], among other things, it was proved that the function ln((sinh x)/x) ln cosh x

(7.21)


is decreasing on (−∞, 0) and increasing on (0, ∞) with range following double inequality was inferred: 3 sinh x sinh x < cosh x < , x 6= 0. x x

29

1 3, 1

. From this, the

(7.22)

It was also mentioned that the inequality (7.20) can be proved directly by applying Lemma 2.1 for f (a) = g(a) = 0 or f (b) = g(b) = 0 to the function (cosh x)−1/3 sinh x . (7.23) x The inequality (7.20) was recovered in [164, Lemma 3]. The inequality (7.20) was refined in [139] as follows: For x 6= 0, the inequality λ sinh x λ λ > cosh x − + 1 (7.24) x 3 3 holds if and only if λ < 0 or λ ≥ 75 and reverses if and only if 0 < λ ≤ 1. The inequality (7.20) was generalized in [7] to modified Bessel functions. Remark 7.1. In [80], it was proved that the inequality 3 sin x > cos x (7.25) x is valid for x ∈ 0, π2 and the exponent 3 is the best possible. See also [79, pp. 238–240]. In [4, pp. 806–807], it was point out that the inequality (7.25) can also be proved directly by Lemma 2.1 for f (a) = g(a) = 0 or f (b) = g(b) = 0 by considering the quotient (cos x)−1/3 sin x (7.26) x and that the inequality (7.25) is the special case g(x) > g(0) = 3 for x on 0, π2 , where ln cos x g(x) = . (7.27) ln((sin x)/x) The inequality (7.25) was refined in [139] as follows: For x ∈ 0, π2 , the inequality λ λ sin x λ (7.28) > cos x − + 1 x 3 3 holds if and only if λ < 0 or λ ≥ λ0 = 1.420 · · · and reverses if and only if 0 < λ ≤ 57 , λ where λ0 satisfies λ3 + π2 − 1 = 0. 7.6. Oppenheim’s problem. Considering inequalities stated in Section 1.7 and Section 7.3, it is natural to ask the following problems: (1) What are the best possible positive constants a, b, c, r and α, β, γ, λ such that sin x α + β cosγ (λx) ≤ ≤ a + b cosc (rx) (7.29) x for − π2 ≤ x ≤ π2 with x 6= 0 and α + β coshγ (λx) ≤

sinh x ≤ a + b coshc (rx) x

(7.30)

30


for −∞ < x < ∞ with x 6= 0 hold respectively? (2) What about the analogues of Bessel functions or other special functions? These problems are similar to Oppenheim’s problem which has been investigated in [7, 12, 20, 98, 162]. 7.7. Some inequalities of Bessel functions. For more information on inequalities of Bessel functions and some other special functions, please refer to [8, 9, 15, 16, 19] and related references therein. 8. Wilker-Anglesio’s inequality and its generalizations 8.1. Wilker’s inequality and generalizations. In [132], J. B. Wilker proved 2 sin x tan x + >2 (8.1) x x and proposed that there exists a largest constant c such that 2 sin x tan x + > 2 + cx3 tan x x x

(8.2)

for 0 < x < π2 . In recent years, Wilker’s inequality (8.1) has been proved once and again in papers such as [4, 30, 39, 73, 127, 134, 148, 161]. In [147], the inequality (8.1) was generalized as: If q > 0 or q ≤ min −1, − µλ , then p q λ sin x µ tan x + >1 (8.3) µ+λ x µ+λ x holds for 0 < x < π2 , where λ > 0, µ > 0 and p ≤ 2qµ λ . As an application of the inequality (8.3), an inequality posed as an open problem in [125] was solved and improved. In [164], the inequality (8.1) was generalized as 2 sinh x tanh x >2 (8.4) + x x for x 6= 0, which and (8.1) were further extended and refined in [139, 169] as 2p p 2p p tanh x x x sinh x + > + > 2, x 6= 0 (8.5) x x sinh x tanh x and

sin x x

2p

+

tan x x

p

>

x sin x

2p

+

x tan x

p > 2,

02+ 45 x x π 4 8 for 0 < x < π2 . The constants 45 and π2 in the inequality (8.7) are the best possible. In [40, 41, 50, 153], several proofs of Wilker-Anglesio’s inequality (8.7) were given. In [90], a new proof of the inequality (8.7) was provided by using Lemma 2.1 and compared with [50]. 2 x In [56, 148, 149], three lower bounds for sinx x + tan x − 2 were presented, but 2 4 3 they are weaker than π x tan x in (8.7). In [130, 131], the following Wilker type inequality was obtained: 2 2 3 x 2 16 x 2 + x sin x < + . (9.13) 5 80 4 − x2 − x3 0 Numerical computation shows that the lower bound in (9.7) is better than those in (9.12) and (9.13). In [151], by directly proving the inequality (9.6) and √ √ √ 1 8 2 −9 2 −1 2 5−4 2 2 1 √ ≤ + x + x (1 − x) √ + x , (9.14) 2 2 8 8 2 − 10 4 − x2 − x3 the inequality (9.7) and an improved upper bound in (9.4), √ Z 1 79 2 1 √ dx < + , 2 3 192 10 4−x −x 0 were obtained. In [101], by considering an auxiliary function √ √ 1 8 2 −9 1 1− 2 2 √ x + αx2 (1 − x) √ +x − + 2 2 8 2 − 10 4 − x2 − x3

(9.15)

(9.16)

on [0, 1], inequalities (9.14) and √ √ √ 1 2 − 1 2 1137 4 2 − 5 8 2 −9 1 √ √ (1 − x) √ x − ≥ + + x (9.17) 2 2 64 64 − 39 2 8 2 − 10 4 − x2 − x3 were demonstrated to be sharp, and then, by integrating on both sides of (9.14), the inequality (9.15) was recovered. 9.2. Estimates for the second kind of complete elliptic integrals. In [49], by discussing p p 4 p π 1 + k 2 cos2 t − 1 + k 2 + 2 1 + k 2 − 1 t2 + θ −t t (9.18) π 2 or p p 2 p π 1 + k 2 cos2 t − 1 + k 2 + 1 + k2 − 1 t + β −t t (9.19) π 2 on 0, π2 , where θ and β are undetermined constants, the inequality π 8 p − 2 1 + k2 − 1 t −t ≤ π 2 p p 4 p 1 + k 2 cos2 t − 1 + k2 − 2 1 + k 2 − 1 t2 ≤ 0 (9.20) π π 2 for t ∈ 0, 2 was obtained, where k 2 = ab 2 − 1 and a, b > 0. Integrating (9.20) yields Z π/2 p π π (2a + b) < a2 sin2 t + b2 cos2 t d t ≤ (a + 2b) (9.21) 6 6 0

34


for b > a. When b ≥ 7a, the right-hand side of the inequality (9.21) is stronger than the well-known result Z π/2 p π πp 2 a2 sin2 t + b2 cos2 t d t ≤ 2(a + b2 ) (9.22) (a + b) ≤ 4 4 0 which can be obtained by using some properties of definite integral or by applying the well-known Hermite-Hadamard double integral inequality for convex functions to the integral in question. Remark 9.1. By employing Lemma 2.1, some inequalities for complete elliptic integrals, including the tighter upper bound for the elliptic integral of the second kind, were obtained in [3]. Remark 9.2. It is worthwhile to point out that some inequalities for bounding complete elliptic integrals of the first and second kinds are presented in [47, 48]. 9.3. Inequalities for the remainder of power series expansion of ex . In [51, 92], by considering the auxiliary function ex − Sn (x) − αn xn+1 + θ(b − x)xn+1 b

for 0 ≤ x ≤ b ∈ (0, ∞), where α−1 = e and αn = inequalities of the reminder Rn (x) = ex −

1 b

αn−1 −

(9.23) 1 n!

, the following

n X xk k=0

k!

(9.24)

for n ≥ 0 and x ∈ [0, ∞) were established: n + 2 − (n + 1)x n+1 x n + 1 + ex n+1 ex x e ≤ Rn (x) ≤ x ≤ xn+1 , (n + 2)! (n + 2)! (n + 1)! k (n + 2)! Rn (x) ≤ xk Rn−k (x) + xn+1 , 0 ≤ k ≤ n (n − k + 2)! (n − k + 2)!

(9.25) (9.26)

and, for n ≥ k ≥ 1, xk Rn−k (x) ≤

kxn+1 ex n! − (n − k + 2)(n + 1)! − Rn (x). (n + 1)(n − k + 2)! (n − k + 2)!

(9.27)

10. Estimates and inequalities for complete elliptic integrals In this section, we continue to recite some estimates and inequalities for complete elliptic integrals and their new developments in recent years. 10.1. Inequalities between three kinds of complete elliptic integrals. By using Tchebycheff’s integral inequality [79, p. 39, Theorem 9], the following inequalities between three kinds of complete elliptic integrals were derived in [110]: π ln((1 + k)/(1 − k)) π arcsin k < K(k) < ; 2k 4k 2 4 16 − 4k − 3k E(k) < K(k); 4(4 + k 2 ) k2 h K(k) < 1 + II(k, h), −1 < h < 0 or h > > 0; 2 2 − 3k 2

(10.1) (10.2) (10.3)


35

π2 II(k, h)E(k) > √ , −2 < 2h < k 2 ; 4 1+h 16 − 28k 2 + 9k 4 2 E(k) ≥ K(k), k 2 ≤ . 4(4 − 5k 2 ) 3

(10.4) (10.5) 2

k For 0 < 2h < k 2 , the inequality (10.3) is reversed. For h > 2−3k 2 > 0, the inequality (10.4) is reversed. As concrete examples, the following estimates of the complete elliptic integrals are also deduced in [110]: √ −1/2 Z π/2 sin2 x π ln(1 + 2 ) π2 √ √ < 1− dx < , (10.6) 2 4 2 2 0 −1 Z π/2 π(ln 3 − ln 2) cos x dx < , (10.7) 1+ 2 2 0 −1 −1 Z π/2 Z π sin x π ln 2 cos x 1− dx = dx > . (10.8) 1+ 2 2 2 0 π/2

These results are better than those in [66, p. 607]. 10.2. Carlson-Vuorinen’s inequality. In [24], the following inequality was proposed: Z dθ 2 π/2 ln b − ln a p ≤ . (10.9) 2 2 2 2 π 0 b−a a cos θ + b sin θ Equality holds if and only if a = b. The inequality (10.9) was recovered in [22, Theorem 4]. There are two natural questions on bounding the complete elliptic integral in (10.9) to ask: (1) What are the best constants β > α > 0 such that the inequality α β Z ln b − ln a 2 π/2 ln b − ln a dθ p ≤ ≤ (10.10) b−a π 0 b−a a2 cos2 θ + b2 sin2 θ holds for all positive numbers a and b with a 6= b? (2) Is the lower bound for (10.9) the reciprocal of the identric mean 1/(b−a) 1 bb I(a, b) = e aa

(10.11)

for positive numbers a and b with a 6= b? Since the complete elliptic integral in (10.9) tends to infinity as the ratio ab for a > b > 0 tends to zero, so we think that the former question is more significant. For more information on the origin, refinements, extensions and generalizations of the inequality (10.9), please refer to [83, 117, 118] and closely related references therein. 10.3. Some recent results of elliptic integrals. The double inequality (10.1) was strengthened in [59, Theorem 4.1] and [60, (1.13)]. It was pointed in [13] that the right-hand side inequality in (10.1) is a recovery of [6, Theorem 3.10]. In [13], the inequality (10.1) was also generalized to the case of generalized complete elliptic integrals by the same method as in [97, 110].

36


Some tighter inequalities than inequalities (10.2) and (10.5) were contained in [60, (3.21)]. The elliptic integral appeared in (10.6) is K √12 which can found in [21]. In [14], some of the results in [13] were further improved. Acknowledgements. The authors express their gratitude to the anonymous referees for many helpful comments and valuable suggestions on this paper. ´ ad Baricz in Romania and many mathematicians in The authors appreciate Arp´ China, such as Chao-Ping Chen, Wei-Dong Jiang, Jian-Lin Li, Shan-He Wu, XiaoMing Zhang and Ling Zhu, for their valuable comments on this manuscript and timely supply of several newly published articles of their own. This paper was ever revised while the first author was visiting the RGMIA, School of Computer Science and Mathematics, Victoria University, Australia between March 2008 and February 2009. Therefore, the first author expresses many thanks to local members of the RGMIA such as Professors Pietro Cerone and Sever S. Dragomir for their invitation and hospitality throughout the period. References [1] U. Abel and D. Caccia, A sharpening of Jordan’s inequality, Amer. Math. Monthly 93 (1986), no. 7, 568–569. 4 [2] R. P. Agarwal, Y.-H. Kim and S. K. Sen, A new refined Jordan’s inequality and its application, Math. Inequal. Appl. 12 (2009), no. 2, 255–264. 15 [3] H. Alzer and S.-L. Qiu, Monotonicity theorems and inequalities for complete elliptic integrals, J. Comput. Appl. Math. 172 (2004), no. 2, 289–312; Available online at http: //dx.doi.org/10.1016/j.cam.2004.02.009. 34 [4] G. Anderson, M. Vamanamurthy and M. Vuorinen, Monotonicity rules in calculus, Amer. Math. Monthly 113 (2006), 805–816. 28, 29, 30 [5] G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen, Conformal Invariants, Inequalities, and Quasi-conformal Maps, John Wiley & Sons, New York, 1997. 8 [6] G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen, Functional inequalities for hypergeometric functions and complete elliptic integrals, SIAM J. Math. Anal. 23 (1992), no. 2, 512–524; Available online at http://dx.doi.org/10.1137/0523025. 35 ´ Baricz, Functional inequalities involving Bessel and modified Bessel functions of the first [7] A. kind, Expo. Math. 26 (2008), no. 3, 279–293. 29, 30 ´ Baricz, Functional inequalities involving special functions, J. Math. Anal. Appl. 319 [8] A. (2006), no. 2, 450–459. 30 ´ Baricz, Functional inequalities involving special functions, II, J. Math. Anal. Appl. 327 [9] A. (2007), no. 2, 1202–1213. 30 ´ Baricz, Jordan-type inequalities for generalized Bessel functions, J. Inequal. Pure Appl. [10] A. Math. 9 (2008), no. 2, Art. 39; Available online at http://www.emis.de/journals/JIPAM/ article971.html. 27 ´ Baricz, Redheffer type inequality for Bessel functions, J. Inequal. Pure Appl. Math. [11] A. 8 (2007), no. 1, Art. 11; Available online at http://www.emis.de/journals/JIPAM/ article824.html. 28 ´ Baricz, Some inequalities involving generalized Bessel functions, Math. Inequal. Appl. 10 [12] A (2007), no. 4, 827–842. 27, 28, 30 ´ Baricz, Tur´ [13] A an type inequalities for generalized complete elliptic integrals, Math. Z. 256 (2007), no. 4, 895–911. 35, 36 ´ Baricz, Tur´ [14] A an type inequalities for hypergeometric functions, Proc. Amer. Math. Soc. 136 (2008), no. 9, 3223–3229. 36 ´ Baricz and E. Neuman, Inequalities involving generalized Bessel functions, J. Inequal. [15] A. Pure Appl. Math. 6 (2005), no. 4, Art 126; Available online at http://www.emis.de/ journals/JIPAM/article600.html. 30


37

´ Baricz and E. Neuman, Inequalities involving modified Bessel functions of the first kind II, [16] A. J. Math. Anal. Appl. 332 (2007), no. 1, 265–271. 30 ´ Baricz and J. S´ [17] A. andor, Extensions of the generalized Wilker inequality to Bessel functions, J. Math. Inequal. 2 (2008), no. 3, 397–406. 30 ´ Baricz and S.-H. Wu, Sharp exponential Redheffer-type inequalities for Bessel functions, [18] A. Publ. Math. Debrecen 74 (2009), no. 3-4, 257–278. 12 ´ Baricz and S.-H. Wu, Sharp Jordan-type inequalities for Bessel functions, Publ. Math. [19] A. Debrecen 74 (2009), no. 1-2, 107–126. 28, 30 ´ Baricz and L. Zhu, Extension of Oppenheim’s problem to Bessel functions, J. Inequal. [20] A. Appl. 2007 (2007), Article ID 82038, 7 pages; Available onine at http://dx.doi.org/10. 1155/2007/82038. 5, 30 [21] J. M. Borwein and P. B. Borwein, Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity, Wiley, New York, 1987. 36 [22] P. Bracken, An arithmetic-geometric mean inequality, Expo. Math. 19 (2001), no. 3, 273–279. 35 [23] P. S. Bullen, A Dictionary of Inequalities, Pitman Monographs and Surveys in Pure and Applied Mathematics 97, Addison Wesley Longman Limited, 1998. 2, 28 [24] B. C. Carlson and M. Vuorinen, An inequality of the AGM and the logarithmic mean, SIAM Rev. 33 (1991), Problem 91-17, 655. 35 [25] W. B. Carver, Extreme parameters in an inequality, Amer. Math. Monthly 65 (1958), no. 2, 206–209. 5 [26] C.-P. Chen and F. Qi, A double inequality for remainder of power series of tangent function, RGMIA Res. Rep. Coll. 5 (2002), Suppl., Art. 2; Available online at http://rgmia.org/ v5(E).php. 6 [27] C.-P. Chen and F. Qi, A double inequality for remainder of power series of tangent function, Tamkang J. Math. 34 (2003), no. 4, 351–355; Available online at http://dx.doi.org/10. 5556/j.tkjm.34.2003.236. 6 [28] C.-P. Chen and F. Qi, Inequalities of some trigonometric functions, RGMIA Res. Rep. Coll. 6 (2003), no. 3, Art. 2, 419–429; Available online at http://rgmia.org/v6n3.php. 5 [29] C.-P. Chen and F. Qi, Inequalities of some trigonometric functions, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 15 (2004), 72–78; Available online at http://dx.doi.org/10. 2298/PETF0415071C. 5 [30] C.-P. Chen and F. Qi, On two new proofs of Wilker’s inequality, G¯ aodˇ eng Sh` uxu´ e Y´ anji¯ u (Studies in College Mathematics) 5 (2002), no. 4, 38–39. (Chinese) 30 [31] C.-P. Chen, J.-W. Zhao, and F. Qi, Three inequalities involving hyperbolically trigonometric functions, Octogon Math. Mag. 12 (2004), no. 2, 592–596. 11, 28 [32] C.-P. Chen, J.-W. Zhao, and F. Qi, Three inequalities involving hyperbolically trigonometric functions, RGMIA Res. Rep. Coll. 6 (2003), no. 3, Art. 4, 437–443; Available online at http://rgmia.org/v6n3.php. 11, 28 [33] J.-X. Cheng and J.-E. Deng, Extension of Jordan inequality, China Science and Technology Information (2008), no. 13, 36. (Chinese) 8 [34] L. Debnath and C.-J. Zhao, New strengthened Jordan’s inequality and its applications, Appl. Math. Lett. 16 (2003), no. 4, 557–560. 14 [35] K. Deng, The noted Jordan’s inequality and its extensions, Xi¯ angt´ an Ku` angy` e Xu´ eyu` an Xu´ eb` ao (Journal of Xiangtan Mining Institute) 10 (1995), no. 4, 60–63. (Chinese) 7, 14 [36] Y.-F. Feng, Proof without words: Jordan’s inequality 2x ≤ sin x ≤ x, 0 ≤ x ≤ π2 , Math. π Mag. 69 (1996), 126. 2 [37] A. M. Fink, Two inequalities, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 6 (1995), 48–49. 5 [38] R. H. Garstang, Bounds on an elliptic integral, Amer. Math. Monthly 94 (1987), no. 6, 556–557. 32 [39] Y.-H. Guo, Two new proofs of Wilker’s inequality, G¯ aodˇ eng Sh` uxu´ e Y´ anji¯ u (Studies in College Mathematics) 9 (2006), no. 4, 79. (Chinese) 30 [40] B.-N. Guo, W. Li and F. Qi, Proofs of Wilker’s inequalities involving trigonometric functions, The 7th International Conference on Nonlinear Functional Analysis and Applications, Chinju, South Korea, August 6-10, 2001; Inequality Theory and Applications, Volume 3, Yeol Je Cho, Jong Kyu Kim, and Sever S. Dragomir (Eds), Nova Science Publishers, Hauppauge, NY, ISBN 1-59033-866-9, 2003, pp. 109–112. 31

38


[41] B.-N. Guo, W. Li, B.-M. Qiao and F. Qi, On new proofs of inequalities involving trigonometric functions, RGMIA Res. Rep. Coll. 3 (2000), no. 1, Art. 15, 167–170; Available online at http://rgmia.org/v3n1.php. 31 [42] B.-N. Guo, Q.-M. Luo, and F. Qi, Monotonicity results and inequalities for the inverse hyperbolic sine function, J. Inequal. Appl. 2013, 2013:536, 6 pages; Available online at http://dx.doi.org/10.1186/1029-242X-2013-536. 5 [43] B.-N. Guo, Q.-M. Luo, and F. Qi, Sharpening and generalizations of Shafer-Fink’s double inequality for the arc sine function, Filomat 27 (2013), no. 2, 261–265; Available online at http://dx.doi.org/10.2298/FIL1302261G. 5 [44] B.-N. Guo and F. Qi, Alternative proofs for inequalities of some trigonometric functions, Internat. J. Math. Ed. Sci. Tech. 39 (2008), no. 3, 384–389; Available online at http: //dx.doi.org/10.1080/00207390701639516. 5 [45] B.-N. Guo and F. Qi, Sharpening and generalizations of Carlson’s double inequality for the arc cosine function, available online at http://arxiv.org/abs/0902.3039. 5 [46] B.-N. Guo and F. Qi, Sharpening and generalizations of Carlson’s inequality for the arc cosine function, Hacet. J. Math. Stat. 39 (2010), no. 3, 403–409. 5 [47] B.-N. Guo and F. Qi, Some bounds for the complete elliptic integrals of the first and second kinds, available online at http://arxiv.org/abs/0905.2787. 34 [48] B.-N. Guo and F. Qi, Some bounds for the complete elliptic integrals of the first and second kinds, Math. Inequal. Appl. 14 (2011), no. 2, 323–334; Available online at http://dx.doi. org/10.7153/mia-14-26. 34 [49] B.-N. Guo, F. Qi and S.-J. Jing, Improvement for the upper bound of a class of elliptic integral, Ji¯ aozu` o Ku` angy` e Xu´ eyu` an Xu´ eb` ao (Journal of Jiaozuo Mining Institute) 14 (1995), no. 6, 125–128. (Chinese) 8, 33 [50] B.-N. Guo, B.-M. Qiao, F. Qi, and W. Li, On new proofs of Wilker’s inequalities involving trigonometric functions, Math. Inequal. Appl. 6 (2003), no. 1, 19–22; Available online at http://dx.doi.org/10.7153/mia-06-02. 31 [51] Q.-D. Hao, On construction of several sharp inequalities for the exponential function ex , Ku` ang Y` e (Mining) (1995), no. 1, 39–42. (Chinese) 34 [52] Q.-D. Hao and B.-N. Guo, A method of finding extremums of composite functions of trigonometric functions, Ku` ang Y` e (Mining) (1993), no. 4, 80–83. (Chinese) 6 [53] A. Hoorfar and F. Qi, Some new bounds for Mathieu’s series, Abstr. Appl. Anal. 2007 (2007), Article ID 94854, 10 pages; Available online at http://dx.doi.org/10.1155/2007/ 94854. 5 [54] A. P. Iuskevici, History of Mathematics in 16th and 17th Centuries, Moskva, 1961. 5 [55] W. Janous, Propblem 1216, Crux Math. 13 (1987), 53. 23 [56] W.-D. Jiang and Y. Hua, Note on Wilker’s inequality and Huygens’s inequality, B` udˇ engsh`ı Y¯ anji¯ u T¯ ongx` un (Communications in Studies on Inequalities) 13 (2006), no. 1, 149–151. 31 [57] W.-D. Jiang and Y. Hua, An improvement and application of Jordan’s inequality, G¯ aodˇ eng Sh` uxu´ e Y´ anji¯ u (Studies in College Mathematics) 9 (2006), no. 5, 60–61. 14 [58] W.-D. Jiang and Y. Hua, Sharpening of Jordan’s inequality and its applications, J. Inequal. Pure Appl. Math. 7 (2006), no. 3, Art. 102; Available online at http://www.emis. de/journals/JIPAM/article719.html. 14 [59] H. Kazi and E. Neuman, Inequalities and bounds for elliptic integrals, J. Approx. Theory 146 (2007), no. 2, 212–226. 35 [60] H. Kazi and E. Neuman, Inequalities and bounds for elliptic integrals II, Contemp. Math. 471 (2008), 127–138. 35, 36 [61] G. Klambauer, Problems and Propositions in Analysis, Marcel Dekker, New York and Basel, 1979. 4 [62] M. S. Klamkin, Solution of Problem 1216, Crux Math. 14 (1988), 120–122. 23 [63] R. Kl´ en, M. Lehtonen and M. Vuorinen, On Jordan type inequalities for hyperbolic functions, available online at http://arxiv.org/abs/0808.1493. 13 [64] H. Kober, Approximation by integral functions in the complex domain, Trans. Amer. Math. Soc. 56 (1944), no. 1, 7–31. 3 [65] J.-C. Kuang, Ch´ angy` ong B` udˇ engsh`ı (Applied Inequalities), H´ un´ an Ji` aoy` u Ch¯ ubˇ an Sh` e (Hunan Education Press), Changsha, China, June 1989. (Chinese) 3, 13 [66] J.-C. Kuang, Ch´ angy` ong B` udˇ engsh`ı (Applied Inequalities), 2nd ed., H´ un´ an Ji` aoy` u Ch¯ ubˇ an Sh` e (Hunan Education Press), Changsha, China, May 1993. (Chinese) 3, 4, 13, 35


39

[67] J.-C. Kuang, Ch´ angy` ong B` udˇ engsh`ı (Applied Inequalities), 3rd ed., Sh¯ and¯ ong K¯ exu´ e J`ısh` u Ch¯ ubˇ an Sh` e (Shandong Science and Technology Press), Ji’nan City, Shandong Province, China, 2004. (Chinese) 2, 3, 4, 13, 28 [68] I. Lazarevi´ c, Sur une in´ egalit´ e de Lochs, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 230–241 (1968), 55–56. 28 [69] M. Lehtonen, Yleistetty konveksisuus, Master Thesis in Mathematics at the University of Turku, written under the supervision of Professor M. Vuorinen, March 2008. 13 [70] J.-L. Li, An identity related to Jordan’s inequality, Intern. J. Math. Math. Sci. 2006 (2006), Article ID 76782, 6 pages; Available online at http://dx.doi.org/10.1155/IJMMS/2006/ 76782. 9 [71] J.-L. Li and Y.-L. Li, On the strengthened Jordan’s inequality, J. Inequal. Appl. 2007 (2007), Article ID 74328, 8 pages; Available online at http://dx.doi.org/10.1155/2007/74328. 4, 9, 10, 11 [72] F.-M. Li and Y.-F. Ma, Notes on Jordan inequality, X¯ı’¯ an Sh´ıy´ ou D` axu´ e Xu´ eb` ao (Z`ır´ an K¯ exu´ e Bˇ an) (Journal of Xi’an Shiyou University (Natural Science Edition)) 19 (2004), no. 5, 85–86. (Chinese) 13 [73] A.-Q. Liu, G. Wang, and W. Li, New proofs of Wilker’s inequalities involving trigonometric functions, Ji¯ aozu` o G¯ ongxu´ eyu` an Xu´ eb` ao (Z`ır´ an K¯ exu´ e Bˇ an) (Journal of Jiaozuo Institute of Technology (Natural Science)) 21 (2002), no. 5, 401–403. (Chinese) 30 [74] Q.-M. Luo, Z.-L. Wei, and F. Qi, Lower and upper bounds of ζ(3), Adv. Stud. Contemp. Math. (Kyungshang) 6 (2003), no. 1, 47–51. 5 [75] Q.-M. Luo, Z.-L. Wei, and F. Qi, Lower and upper bounds of ζ(3), RGMIA Res. Rep. Coll. 4 (2001), no. 4, Art. 7, 565–569; Available online at http://rgmia.org/v4n4.php. 5 [76] B. J. Maleˇsevi´ c, An application of λ-method on Shafer-Fink’s inequality, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 8 (1997), 90–92. 5 [77] A. McD. Mercer, Problem E 2952, Amer. Math. Monthly 89 (1982), no. 6, 424. 4 [78] M. Merkle, Inequalities for the gamma function via convexity, In: P. Cerone, S. S. Dragomir (Eds.), “Advances in Inequalities for Special Functions”, Nova Science Publishers, New York, 2008, 81–100. 13 [79] D. S. Mitrinovi´ c, Analytic Inequalities, Springer-Verlag, 1970. 2, 3, 5, 8, 13, 28, 29, 34 [80] D. S. Mitrinovi´ c and D. D. Adamovi´ c, Sur une in´ egalit´ e´ el´ ementaire o` u interviennent des fonctions trigonom´ etriques, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 143-155 (1965), 23–34. 29 [81] V. N. Murty and G. P. Henderson, Solution of Problem 1365, Crux Math. 15 (1989), 251–256. 23 [82] E. Neuman, Inequalities involving Bessel functions of the first kind, J. Inequal. Pure Appl. Math. 5 (2004), no 4, Art. 94; Available online at http://www.emis.de/journals/JIPAM/ article449.html. 27 [83] E. Neuman and J. S´ andor, On certain means of two arguments and their extensions, Int. J. Math. Math. Sci. 16 (2003), 981–993. 35 [84] D.-W. Niu, Generalizations of Jordan’s Inequality and Applications, Thesis supervised by Professor Feng Qi and submitted for the Master Degree of Science at Henan Polytechnic University in June 2007. (Chinese) 16, 17, 25, 26, 27 [85] D.-W. Niu, J. Cao, and F. Qi, Generalizations of Jordan’s inequality and concerned relations, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 72 (2010), no. 3, 85–98. 26, 27 [86] D.-W. Niu, Z.-H. Huo, J. Cao, and F. Qi, A general refinement of Jordan’s inequality and a refinement of L. Yang’s inequality, Integral Transforms Spec. Funct. 19 (2008), no. 3, 157–164; Available online at http://dx.doi.org/10.1080/10652460701635886. 16 [87] Z.-H. Huo, D.-W. Niu, J. Cao, and F. Qi, A generalization of Jordan’s inequality and an application, Hacet. J. Math. Stat. 40 (2011), 53–61. 16 [88] A. Oppenheim, E1277, Amer. Math. Monthly 64 (1957), no. 6, 504. 5 ¨ [89] A. Y. Ozban, A new refined form of Jordan’s inequality and its applications, Appl. Math. Lett. 19 (2006), no. 2, 155–160. 14 [90] I. Pinelis, L’Hospital rules for monotonicity and the Wilker-Anglesio inequality, Amer. Math. Monthly 111 (2004), no. 10, 905–909; Available online at http://dx.doi.org/10. 2307/4145099. 31

40


[91] J. Prestin, Trigonometric interpolation in H¨ older spaces, J. Approx. Theory 53 (1988), no. 2, 145–154. 4 [92] F. Qi, A method of constructing inequalities about ex , Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 8 (1997), 16–23. 31, 34 [93] F. Qi, Bounds for the ratio of two gamma functions, RGMIA Res. Rep. Coll. 11 (2008), no. 3, Art. 1; Available online at http://rgmia.org/v11n3.php. 24 [94] F. Qi, Bounds for the ratio of two gamma functions, J. Inequal. Appl. 2010 (2010), Article ID 493058, 84 pages; Available online at http://dx.doi.org/10.1155/2010/493058. 24 [95] F. Qi, Extensions and sharpenings of Jordan’s and Kober’s inequality, G¯ ongk¯ e Sh` uxu´ e (Journal of Mathematics for Technology) 12 (1996), no. 4, 98–102. (Chinese) 8, 14, 31 [96] F. Qi, Jordan’s inequality: Refinements, generalizations, applications and related problems, RGMIA Res. Rep. Coll. 9 (2006), no. 3, Art. 12; Available online at http://rgmia.org/ v9n3.php. 1, 28 [97] F. Qi, L.-H. Cui, and S.-L. Xu, Some inequalities constructed by Tchebysheff ’s integral inequality, Math. Inequal. Appl. 2 (1999), no. 4, 517–528; Available online at http://dx. doi.org/10.7153/mia-02-42. 5, 8, 28, 35 [98] F. Qi and B.-N. Guo, A concise proof of Oppenheim’s double inequality relating to the cosine and sine functions, available online at http://arxiv.org/abs/0902.2511. 5, 30 [99] F. Qi and B.-N. Guo, Concise sharpening and generalizations of Shafer’s inequality for the arc sine function, available online at http://arxiv.org/abs/0902.2588. 5 [100] F. Qi and B.-N. Guo, Extensions and sharpenings of the noted Kober’s inequality, Ji¯ aozu` o Ku` angy` e Xu´ eyu` an Xu´ eb` ao (Journal of Jiaozuo Mining Institute) 12 (1993), no. 4, 101–103. (Chinese) 6, 7, 8, 31 [101] F. Qi and B.-N. Guo, Estimate for upper bound of an elliptic integral, Sh` uxu´ e de Sh´ıji` an yˇ u R` ensh´ı (Math. Practice Theory) 26 (1996), no. 3, 285–288. (Chinese) 8, 33 [102] F. Qi and B.-N. Guo, Lower bound of the first eigenvalue for the Laplace operator on compact Riemannian manifold, Chinese Quart. J. Math. 8 (1993), no. 2, 40–49. 6 [103] F. Qi and B.-N. Guo, On generalizations of Jordan’s inequality, M´ eit` an G¯ aodˇ eng Ji` aoy` u (Coal Higher Education), Suppl., November/1993, 32–33. (Chinese) 7, 8, 31 [104] F. Qi and B.-N. Guo, Sharpening and generalizations of Carlson’s inequality for the arc cosine function, available online at http://arxiv.org/abs/0902.3495. 5 [105] F. Qi and B.-N. Guo, Sharpening and generalizations of Shafer’s inequality for the arc sine function, Integral Transforms Spec. Funct. 23 (2012), no. 2, 129–134; Available online at http://dx.doi.org/10.1080/10652469.2011.564578. 5 [106] F. Qi and B.-N. Guo, Sharpening and generalizations of Shafer-Fink’s double inequality for the arc sine function, available online at http://arxiv.org/abs/0902.3036. 5 [107] F. Qi and B.-N. Guo, Sharpening and generalizations of Shafer’s inequality for the arc tangent function, available online at http://arxiv.org/abs/0902.3298. 5 [108] F. Qi and B.-N. Guo, The estimation of inferior bound for an ellipse integral, G¯ ongk¯ e Sh` uxu´ e (Journal of Mathematics for Technology) 10 (1994), no. 1, 87–90. (Chinese) 7, 32 [109] F. Qi and Q.-D. Hao, Refinements and sharpenings of Jordan’s and Kober’s inequality, Mathematics and Informatics Quarterly 8 (1998), no. 3, 116–120. 8, 28, 31 [110] F. Qi and Zh. Huang, Inequalities of the complete elliptic integrals, Tamkang J. Math. 29 (1998), no. 3, 165–169. 34, 35 [111] F. Qi, H.-C. Li, B.-N. Guo and Q.-M. Luo, Inequalities and estimates of the eigenvalue for Laplace operator, Ji¯ aozu` o Ku` angy` e Xu´ eyu` an Xu´ eb` ao (Journal of Jiaozuo Mining Institute) 13 (1994), no. 3, 89–95. (Chinese) 6 [112] F. Qi and Q.-M. Luo, Bounds for the ratio of two gamma functions—From Wendel’s and related inequalities to logarithmically completely monotonic functions, Banach J. Math. Anal. 6 (2012), no. 2, 132–158. 24 [113] F. Qi and Q.-M. Luo, Bounds for the ratio of two gamma functions: from Wendel’s asymptotic relation to Elezovi´ c-Giordano-Peˇ cari´ c’s theorem, J. Inequal. Appl. 2013, 2013:542, 20 pages; Available online at http://dx.doi.org/10.1186/1029-242X-2013-542. 24 [114] F. Qi, Q.-M. Luo, and B.-N. Guo, A simple proof of Oppenheim’s double inequality relating to the cosine and sine functions, J. Math. Inequal. 6 (2012), no. 4, 645–654; Available online at http://dx.doi.org/10.7153/jmi-06-63. 5


41

[115] F. Qi, D.-W. Niu and J. Cao, A general generalization of Jordan’s inequality and a refinement of L. Yang’s inequality, RGMIA Res. Rep. Coll. 10 (2007), Suppl., Art. 3; Available online at http://rgmia.org/v10(E).php. 17 [116] F. Qi, D.-W. Niu, and B.-N. Guo, Refinements, generalizations, and applications of Jordan’s inequality and related problems, J. Inequal. Appl. 2009 (2009), Article ID 271923, 52 pages; Available online at http://dx.doi.org/10.1155/2009/271923. 1 [117] F. Qi and A. Sofo, An alternative and united proof of a double inequality for bounding the arithmetic-geometric mean, available online at http://arxiv.org/abs/0902.2515. 35 [118] F. Qi and A. Sofo, An alternative and united proof of a double inequality for bounding the arithmetic-geometric mean, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 71 (2009), no. 3, 69–76. 35 [119] F. Qi, S.-Q. Zhang, and B.-N. Guo, Sharpening and generalizations of Shafer’s inequality for the arc tangent function, J. Inequal. Appl. 2009 (2009), Article ID 930294, 9 pages; Available online at http://dx.doi.org/10.1155/2009/930294. 5 [120] Th. M. Rassias, Problem E 3111, Amer. Math. Monthly 92 (1985), no. 9, 665. 32 [121] R. Redheffer, Problem 5642, Amer. Math. Monthly 75 (1968), no. 10, 1125. 3 [122] R. Redheffer, Correction, Amer. Math. Monthly 76 (1969), no. 4, 422. 3 [123] J. S´ andor, A note on certain Jordan type inequalities, RGMIA Res. Rep. Coll. 10 (2001), no. 1, Art. 1; Available online at http://rgmia.org/v10n1.php. 13 [124] J. S´ andor, On the concavity of sinx , Octogon Math. Mag. 13 (2005), no. 1, 406–407. 13 x [125] J. S´ andor and M. Bencze, On Huygens’s trigonometric inequality, RGMIA Res. Rep. Coll. 8 (2005), no. 3, Art. 14; Available online at http://rgmia.org/v8n3.php. 5, 30 [126] J. S. Sumner, A. A. Jagers, M. Vowe, and J. Anglesio, Inequalities involving trigonometric functions, Amer. Math. Monthly 98 (1991), no. 3, 264–267. 31 [127] J.-S. Sun, Two simple proof of Wiker’s inequality involving trigonometric functions, G¯ aodˇ eng Sh` uxu´ e Y´ anji¯ u (Studies in College Mathematics) 7 (2004), no. 4, 43. (Chinese) 30 [128] G. Tsintsifas, Problem 1365, Crux Math. 14 (1988), 202. 23 [129] L.-C. Wang, Further extension of the Jordan inequality, D´ axi` an Sh¯if` an Gˇ aodˇ eng Zhu¯ ank¯ e Xu´ exi` ao Xu´ eb` ao (Z`ır´ an K¯ exu´ e Bˇ an) (Journal of Daxian Teachers College (Natural Science Edition)) 14 (2004), no. 5, 3–4. (Chinese) 13 [130] Z.-H. Wang, A new Wilker-type inequality, J. Yinbin Univ. 7 (2007), no. 6, 21–22. (Chinese) 31 [131] Z.-H. Wang and X.-M. Zhang, Introduction to a Wilker type inequality, B` udˇ engsh`ı Y¯ anji¯ u T¯ ongx` un (Communications in Studies on Inequalities) 13 (2006), no. 3, 272–274. (Chinese) 31 [132] J. B. Wilker, Problem E 3306, Amer. Math. Monthly 96 (1989), no. 1, 55. 30 [133] J. P. Williams, A delightful inequality, Amer. Math. Monthly 76 (1969), no. 10, 1153–1154. 3 [134] S.-H. Wu, On extension and refinement of Wilker’s inequality, Rocky Mountains J. Math. 38 (2008), no. 6, 683–687. 30 [135] S.-H. Wu, On generalizations and refinements of Jordan type inequality, Octogon Math. Mag. 12 (2004), no. 1, 267–272. 12 [136] S.-H. Wu, On generalizations and refinements of Jordan type inequality, RGMIA Res. Rep. Coll. 7 (2004), Suppl., Art. 2; Available online at http://rgmia.org/v7(E).php 12 [137] S.-H. Wu, Refinement and generalization of the Jordan’s inequality, Ch´ engd¯ u D` axu´ e Xu´ eb` ao (Z`ır´ an K¯ exu´ e Bˇ an) (Journal of Chengdu University (Natural Science)) 23 (2004), no. 2, 37–40. (Chinese) 12 [138] S.-H. Wu, Sharpness and generalization of Jordan’s inequality and its application, Taiwanese J. Math. 12 (2008), no. 2, 325–336. 20 ´ Baricz, Generalizations of Mitrinovi´ [139] Sh-H. Wu and A. c, Adamovi´ c and Lazarevi´ c’s inequalities and their applications, Publ. Math. Debrecen 75 (2009), no. 3-4, 447–458. 5, 29, 30 [140] Sh-H. Wu and L. Debnath, A new generalized and sharp version of Jordan’s inequality and its applications to the improvement of the Yang Le inequality, Appl. Math. Lett. 19 (2006), 1378–1384. 20 [141] Sh-H. Wu and L. Debnath, A new generalized and sharp version of Jordan’s inequality and its applications to the improvement of the Yang Le inequality, II, Appl. Math. Lett. 20 (2007), no. 5, 532–538. 21, 22

42


[142] Sh-H. Wu and L. Debnath, Generalizations of a parameterized Jordan-type inequality, Janouss inequality and Tsintsifass inequality, Appl. Math. Lett. 22 (2009), no. 1, 130–135. 22 [143] Sh-H. Wu and L. Debnath, Jordan-type inequalitities for differentiable functions and their applications, Appl. Math. Lett. 21 (2008), no. 8, 803–809. 24 [144] S.-H. Wu and H. M. Srivastava, A further refinment of a Jordan type inequalty and its application, Appl. Math. Comput. 197 (2008), no. 2, 914–923. 23, 24 [145] S.-H. Wu, H. M. Srivastava and L. Debnath, Some refined families of Jordan type inequalities and their applications, Integral Transforms Spec. Funct. 19 (2008), 183–193. 24 [146] S.-H. Wu and H. M. Srivastava, A further refinement of Wilker’s inequality, Integral Transforms Spec. Funct. 19 (2008), no. 10, 757–765. 31 [147] S.-H. Wu and H. M. Srivastava, A weighted and exponential generalization of Wilker’s inequality and its applications, Integral Transforms Spec. Funct. 18 (2007), no. 8, 529–535. 30 [148] Y.-F. Wu and X.-S. Xu, Simple proofs and a refinement of Wilker’s inequality, T´ ongl´ıng Xu´ eyu` an Xu´ eb` ao (Journal of Tongling College) 5 (2006), no. 2, 72–88. (Chinese) 30, 31 ¯ a Sh¯if` [149] S.-C. Yang, A sharpening of Wilker inequality, Ab` an G¯ aodˇ eng Zhu¯ ank¯ e Xu´ exi` ao Xu´ eb` ao (Journal of Aba Teachers College) (2003), no. 3, 104–105. (Chinese) 31 [150] L. Yang, Zh´ı F¯ enb` u Lˇıl` un J´ıq´ı X¯ın Y´ anji¯ u (Theory of Value Distribution of Functions and New Research), Science Press, Beijing, 1982. (Chinese) 13 [151] L.-Q. Yu, F. Qi and B.-N. Guo, Estimates for upper and lower bounds of a complete elliptic integral, Ku` ang Y` e (Mining) (1995), no. 1, 35–38. (Chinese) 8, 33 [152] X.-H. Zhang, G.-D. Wang, Y.-M. Chu, Extensions and sharpenings of Jordan’s and Kober’s inequalities, J. Inequal. Pure Appl. Math. 7 (2006), no. 2, Art. 63; Available online at http://www.emis.de/journals/JIPAM/article680.html. 3, 9 [153] L. Zhang and L. Zhu, A new elementary proof of Wilker’s inequalities, Math. Inequal. Appl. 11 (2008), no. 1, 149–151. 31 [154] C.-J. Zhao, Generalization and strengthening of the Yang Le inequality, Sh` uxu´ e de Sh´ıji` an yˇ u R` ensh´ı (Math. Practice Theory) 30 (2000), no. 4, 493–497 (Chinese). 13, 14 [155] C.-J. Zhao, On several new inequalities, Chinese Quart. J. Math. 16 (2001), no. 2, 41–45. 13 [156] C.-J. Zhao and L. Debnath, On generalizations of L. Yang’s inequality, J. Inequal. Pure Appl. Math. 3 (2002), no. 4, Art. 56; Available online at http://www.emis.de/journals/ JIPAM/article208.html. 13, 14 [157] J.-L. Zhao, Q.-M. Luo, B.-N. Guo, and F. Qi, Remarks on inequalities for the tangent function, Hacet. J. Math. Stat. 41 (2012), no. 4, 499–506. 6 [158] J.-L. Zhao, C.-F. Wei, B.-N. Guo, and F. Qi, Sharpening and generalizations of Carlson’s double inequality for the arc cosine function, Hacet. J. Math. Stat. 41 (2012), no. 2, 201–209. 5 [159] L. Zhu, A general form of Jordan’s inequalities and its applications, Math. Inequal. Appl. 11 (2008), no. 4, 655–665. 19, 20, 25 [160] L. Zhu, A general refinement of Jordan-type inequality, Comput. Math. Appl. 55 (2008) 2498–2505. 16, 18, 19 [161] L. Zhu, A new simple proof of Wilker’s inequality, Math. Inequal. Appl. 8 (2005), no. 4, 749–750. 30 [162] L. Zhu, A solution of a problem of Oppeheim, Math. Inequal. Appl. 10 (2007), no. 1, 57–61. 5, 30 [163] L. Zhu, General forms of Jordan and Yang Le inequality, Appl. Math. Lett. 22 (2009), no. 2, 236–241. 18, 19 [164] L. Zhu, On Wilker-type inequalities, Math. Inequal. Appl. 10 (2007), no. 4, 727–731. 29, 30 [165] L. Zhu, Sharpening of Jordan’s inequalities and its applications, Math. Inequal. Appl. 9 (2006), no. 1, 103–106. 18 [166] L. Zhu, Sharpening Jordan’s inequality and the Yang Le inequality, Appl. Math. Lett. 19 (2006), no. 3, 240–243. 15 [167] L. Zhu, Sharpening Jordan’s inequality and the Yang Le inequality, II, Appl. Math. Lett. 19 (2006), no. 9, 990–994. 15 [168] L. Zhu, Sharpening Redheffer-type inequalities for circular functions, Appl. Math. Lett. 22 (2009), no. 5, 743–748. 12


43

[169] L. Zhu, Some new Wilker type inequalities for circular and hyperbolic functions, Abstr. Appl. Anal. 2009 (2009), Article ID 485842, 9 pages; Available online at http://dx.doi. org/10.1155/2009/485842. 30 [170] L. Zhu and J. Sun, Six new Redheffer-type inequalities for circular and hyperbolic functions, Comput. Math. Appl. 56 (2008), no. 2, 522–529. 11 (F. Qi) Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, 300160, China E-mail address: [email protected], [email protected], [email protected] URL: http://qifeng618.spaces.live.com (D.-W. Niu) College of Information and Business, Zhongyuan University of Technology, Zhengzhou City, Henan Province, 450007, China E-mail address: [email protected], [email protected], [email protected] (B.-N. Guo) School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China E-mail address: [email protected], [email protected] URL: http://guobaini.spaces.live.com