On average values of convex functions

Aequat. Math. 86 (2013), 137–154 c Springer Basel 2013 0001-9054/13/010137-18 published online January 4, 2013 DOI 10.1007/s00010-012-0181-7

Aequationes Mathematicae

On average values of convex functions ˇaric ´, I. Peric ´ and G. Roqia J. Pec

Abstract. The aim of this paper is to give an extension of an inequality proved by Wulbert (Math Comput Model 37:1383–1391, 2003, Lemma 2.5) and to define Stolarsky type means as an application of this inequality. Further, we discuss some properties of averages of a continuous convex function, some consequences of a double inequality given by Wulbert (Math Comput Model 37:1383–1391, 2003, Theorem 3.3) and obtain improvement results of Wulbert (Math Comput Model 37:1383–1391, 2003, Corollary 4.3). Mathematics Subject Classification (1991). Primary 26D15; Secondary 26D99. Keywords. Jensen’s inequality, Favard’s inequality, log-convex function, exponential convexity, the Stolarsky quotients.

1. Introduction The Jensen inequality for convex functions is the most important inequality in mathematics and statistics and has many versions. The following version of the Jensen inequality provides the lower bound of the average value of ψ(f (t)) on [a, b] , that is for an integrable function f and a convex function ψ: ⎛ ⎞ b b 1 1 ψ(f (t))dt ≥ ψ ⎝ f (t)dt⎠ . (1) b−a b−a a

a

An upper bound is given by Favard (see [9–11,14] ) 1 2f

2f 0

1 ψ(u)du ≥ b−a

b ψ(f (t))dt,

(2)

a

where f is a non-negative continuous concave function on [a, b], ψ is a convex b 1 function on [0, 2f ] and f = b−a f (t)dt. a

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ˇaric ´ et al. J. Pec

Karlin and Studden improve (2) (see [6,9–11]): 1 2f − 2c

2f −c

c

1 ψ(u)du ≥ b−a

b ψ(f (t))dt,

(3)

a

fmin ≥ c ≥ 0, fmin is the minimum of f on [a, b]. Further, a more general upper bound and the sharp lower bound of the average value of ψ(f (t)) on [a, b] was discovered by Wulbert [14, Theorem 3.3], [14, Theorem 3.3]: Theorem 1.1. If f is concave on [a, b] and ψ is convex on the intervals of integration, then 1 2f − 2fmin

2f−f min

fmin

1 ψ(u)du ≥ b−a

b ψ(f (t))dt ≥ a

1 2fmax − 2f

fmax

ψ(u)du 2f −fmax

(4) holds, where fmax is the maximum of f on [a, b]. If f is convex then the inequality signs in (4) are reversed. The following result is proved in the same paper: Theorem 1.2. If ψ is a nonlinear, convex function on an open interval I, a, b ∈ I, a < b, then 1 2x + b − a

b+x ψ(t)dt > a−x

1 b−a

b ψ(t)dt a

holds for x > 0, (a − x, b + x) ⊆ I. The object of this paper is to study functions defined by averages of a convex function, to extend and improve some inequalities given in [14], and to study associated functionals on the cone of convex functions. We apply a method (given in [8]) of producing n-exponentially convex functions and also log-convex functions using the defined functionals. These functions are used to define Stolarsky type quotients, Lyapunov’s inequality which is used in improving results in [14, Corollary 4.2] and to make a connection with already proved results in [10,11]. In Sect. 2, we study some properties of functions which are defined by averages of a convex function and give an extension of Theorem 1.2. The functionals are obtained using Theorems 1.1 and 1.2. Cauchy type mean values theorems are proved. In Sect. 3, we give a general method of defining exponentially convex functions using the functionals defined in Sect. 2. Also the log-convexity of these functions gives us Lypaunov’s inequality and the monotonicity of associated Stolarsky type quotients. In Sect. 4 we apply results from Sect. 3 on some

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families of exponentially convex functions. The explicit expression of Stolarsky type quotients are calculated and using mean-values theorems proved in Sect. 2, Stolarsky type means are obtained. Using Lypaunov’s inequality, we obtain improvements of [14, Corollary 4.2]. For the convenience of the reader, we state here, main definitions and properties of convex functions, needed for the next results. By [x0 , x1 , . . . , xn ; f ] we denote the divided difference of order n ∈ N of a function f at nodes x0 , x1 , . . . , xn . C1 : The function ψ is convex on an interval I if and only if [u, x, y; ψ] =

[x, y; ψ] − [u, x; ψ] ≥0 y−u

(5)

[u, x; ψ] = (ψ(x) − ψ(u))/(x − u) and [y, x; ψ] = (ψ(x) − ψ(y))/(x − y) holds for distinct x, y, u ∈ I (see [5,9,12]). C2 : If the function ψ is differentiable on an interval I, then ψ is convex on I if and only if [x, x, y; ψ] =

ψ(y) − Sx (y) ≥0 (y − x)2

(6)

holds for Sx (y) = ψ(x) + ψ (x)(y − x) and distinct x, y ∈ I (see [5]). C3 : If ψ exists on an interval I, then ψ is convex on I if and only if ψ (x) ≥0 (7) 2 holds for any x ∈ I (see [5,9,12]). C4 : If ψ is continuous on an interval I, then ψ is convex if and only if [x, x, x; ψ] =

1 ψ(x) + ψ(y) ≥ 2 y−x

y ψ(t)dt

(8)

x

is valid for any x, y ∈ I (see [9,12]). The authors were inspired by the proof of an inequality given in [14, Lemma 2.5], which is used in proving Theorem 1.1.

2. Main results For a, b ∈ R with a ≤ b, d ∈ R+ and a continuous function ψ defined on [a − d, b + d]. Define Ψ and Φ on [a − b − d, d] by ⎧ b+x 1 ψ(t)dt, x = a−b ⎨ 2x+b−a 2 , a−x (9) Ψ(x) = ⎩ a+b a−b ψ( 2 ), x= 2

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and

⎧

b+x 1 1 a+b ⎪ ψ(t)dt − ψ( ) , x = ⎨ 2x+b−a 2x+b−a a−x 2 Φ(x) =

⎪ ⎩

a−b 2 ,

(10) 0,

x=

a−b 2

We will explore some properties of Ψ and Φ and in the sequel we will extend the inequality given in Theorem 1.2. The following lemma is useful in studying averages of functions (see [3]). Lemma 2.1. Let ψ be a continuous function on an interval I with a non-empty interior. If ψ is convex on I, then the function ψ defined by ⎧ 1 y ⎨ y−x x ψ(t)dt, x, y ∈ I, x = y, F (x, y) = ⎩ ψ(x), x = y, is convex on I 2 . Furthermore, for xi , yi ∈ I, i = 1, . . . , n and non-negative real n weights pi , i = 1, . . . , n such that i=1 pi = 1, the following holds n n n pi F (xi , yi ) ≥ F pi xi , pi y i . (11) i=1

i=1

i=1

Theorem 2.2. If ψ is continuous on [a − d, b + d], then the function Ψ is symmetric about the line x = a−b 2 , that is Ψ(a − b − x) = Ψ(x),

x ∈ [a − b − d, d].

Moreover, if ψ is convex on [a − d, b + d], then the following statements hold. (i)

a−b The function Ψ is decreasing on [a−b−d, a−b 2 ] and increasing on [ 2 , d]. a−b Furthermore, Ψ has a minimum value at x = 2 , that is for x ∈ a−b [a − b − d, a−b 2 ) ∪ ( 2 , d]:

1 2x + b − a (ii)

b+x a+b ψ(t)dt ≥ ψ . 2 a−x

The function Ψ is a convex function on [a − b − d, d].

Proof. By simple computation, we have −1 Ψ(a − b − x) = 2x + b − a = Ψ(x).

a−x

ψ(t)dt b+x

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141

Differentiating (9), we have ⎡ ⎤ b+x ψ(a − x) + ψ(b + x) 1 2 ⎣ − Ψ (x) = f (t)dt⎦ . (2x + b − a) 2 2x + b − a a−x

By (8), Ψ (x) ≤ 0 for x ∈ [a−b−d, and Ψ (x) ≥ 0 when x ∈ [ a−b 2 , d]. Consequently Ψ has a minimum value at x = a−b , that is 2 a−b 2 ]

b+x a+b ψ(t)dt ≥ ψ 2

1 2x + b − a

(ii)

a−x

a−b − b − d, a−b 2 ) ∪ ( 2 , d]. ∈ [a − d, b − xi , b + xi

for x ∈ [a + d]) and non-negative real weights Choose a pi , i = 1, . . . , n, with i=1n pi = 1. By substituting xi → a − xi and yi → b + xi in (11), we obtain n n pi Ψ(xi ) ≥ Ψ pi xi . (12) i=1

i=1

By the discrete Jensen inequality, Ψ is convex on [a − b − d, d]. The second Proof of (ii): Using an approximation argument (for example Bernstein polynomials), without loss of generality we can assume that ψ has a continuous second derivative on [a − b − d, d]. We have 8 ψ(a − x) + ψ(b + x) Ψ (x) = Ψ(x) − 2 (2x + b − a) 2 2x + b − a + (ψ (b + x) − ψ (a − x)) . 8 The following identity is given in [3]. If ψ : [a, b] → R (a, b ∈ R ) has a continuous second derivative, then 1 b−a

b ψ(t)dt − a

1 = 2(b − a)

ψ(a) + ψ(b) b − a + [ψ (b) − ψ (a)] 2 8

b

a+b t− 2

2

ψ (t)dt.

(13)

a

Therefore

Ψ (x) =

b+x

4 (2x + b − a)

t−

3

a+b 2

2

ψ (t)dt.

(14)

a−x

Since ψ is convex on [a − d, b + d], Ψ is convex on [a − b − d, d].

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a−b Remark 2.3. Since [0, d] ⊆ [ a−b 2 , d] and Ψ is increasing on [ 2 , d], for any x ∈ [0, d], the following inequality holds

1 2x + b − a

b+x ψ(t)dt ≥ a−x

1 b−a

b ψ(t)dt.

(15)

a

This is the same inequality as that given in Theorem 1.2 with a different proof. Remark 2.4. It is interesting to note that, the reverse implication of (i) does not hold, consider a = b = 0 and d = ∞ and define ψ : R → R by t2 , |t| ≤ 1, ψ(t) = 2 1 − 6t2 + 23 |t|, |t| ≥ 1. The corresponding x2 Ψ(x) =

6 , − 6x1 2 +

x 3

−

1 3x ,

1 ≥ x ≥ 0, x ≥ 1,

and Ψ(x) = Ψ(−x) for 0 ≥ x. It is observed that Ψ is increasing on (0, ∞), but ψ is convex only on (−1, 1). Similarly, the convexity of Ψ on [a − b − d, d] is not sufficient for the convexity of ψ on [a − d, b + d]. For a = b = 0, d = 1, define ψ : [−1, 1] → R by ψ(t) =

t4 3t2 − . 10 12

From (9), the definition of Ψ : [−1, 1] → R reduces to Ψ(x) =

x2 x4 − . 10 60

It is easy to verify that Ψ is convex on [−1, 1], but ψ is not convex on the whole interval [−1, 1]. Theorem 2.5. Suppose that ψ is continuous on [a − d, b + d], then Φ is anti symmetric about the line x = a−b 2 , that is Φ (a − b − x) = −Φ(x),

x ∈ [a − b − d, d].

Furthermore, if ψ is convex on [a−d, b−d], then Φ is increasing on [a−b−d, d]. Proof. From (10) and part (i) of Theorem 2.2, we have Ψ(x) − ψ( a+b 2 ) , a − b − 2x = −Φ (x) .

Φ(a − b − x) =

x =

a−b 2

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This shows that Φ is antisymmetric about the line x = b−a 2 . We apply elementary calculus on (10) to obtain a+b 4 Φ (x) = ψ(a − x) + ψ(b + x) + 2ψ − 4Ψ(x) . 2 2 (2x + b − a) (16) By replacing x → a − x, y → ψ(a − x) + ψ and substituting x → ψ

a+b 2 ,

a+b 2

a+b 2

in (8), we have

a+b 2

a+b

≥

2

4 2x + b − a

ψ(t)dt

(17)

b+x ψ(t)dt.

(18)

a−x

y → b + x in (8), we have

4 + ψ(b + x) ≥ 2x + b − a

a+b 2

Adding (17) and (18), we conclude that Φ is increasing on [a − b − d, d].

For every x ∈ [0, d] define a linear functional W1 on the real vector space of continuous functions on [a − d, b + d] by: 1 W1 (ψ) = 2x + b − a

b+x ψ(t)dt − a−x

1 b−a

b ψ(t)dt,

(19)

a

which is non-negative on the cone of convex function due to (15). Consider a concave function f on [a, b] (−∞ < a < b < ∞) and a continuous function ψ on [fmin , 2f − fmin ]. By taking the differences of double inequality (4), define linear functionals on the real vector space of continuous functions on [fmin , 2f − fmin ] by: 1 W2 (ψ) = 2f − 2fmin

W3 (ψ) =

1 b−a

2f −fmin

fmin

b ψ(f (t))dt − a

1 ψ(u)du − b−a 1 2fmax − 2f

b ψ(f (t))dt

(20)

ψ(u)du.

(21)

a

fmax

2f −fmax

Clearly, Wi , i = 2, 3 are non-negative on the cone of convex functions by Theorem 1.1. In the later text, set I = [0, d] for W1 , I = [fmin , 2f − fmin ] for W2 and W3 as abbreviations. Lagrangian type mean value theorems are essential to produce Cauchy type means from Stolarsky type quotients.

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Analogously to the proofs of Theorem 2.5 and Theorem 2.6 in [7], we prove the following theorems, respectively. Theorem 2.6. If ψ ∈ C 2 (I) and the linear functionals Wi , i = 1, 2, 3 are defined as in (19), (20) and (21), respectively. Then there exists ξ ∈ I such that Wi (ψ) =

ψ (ξ) Wi (e2 ) 2

(22)

where e2 (x) = x2 . Theorem 2.7. If Wi , i = 1, 2, 3 are defined as in (19), (20) and (21), respectively and ψ, φ ∈ C 2 (I) such that φ does not vanish on I. Then there exists ξ ∈ I such that Wi (ψ) ψ (ξ) = , (23) φ (ξ) Wi (φ) assuming the denominator on the right hand side is non-zero.

Remark 2.8. Suppose that the inverse of ψφ exists, then Theorem (2.7) enables us to define various types of means for each i = 1, 2, 3, that is −1 ψ Wi (ψ) ξ= . φ Wi (φ) The number ξ ∈ I, we call Cauchy mean.

3. Exponential convexity of the Wulbert inequality differences Exponentially convex functions were discovered by Bernstein in [2] as a subclass of convex functions (also subclass of log-convex functions) on a given open interval. General results about exponential convexity can be found in [1] and [4]. The notion of n-exponential convexity is introduced in [8]. If not specified, in the sequel, J stands for an open interval in R. Definition 1. For fixed n ∈ N, a function ψ : J → R is n-exponentially convex in the Jensen sense on J if n pi + pj ξi ξj ψ ≥0 2 i,j=1 holds for all choices of ξi ∈ R, pi ∈ J, i = 1, . . . , n. A function ψ : J → R is n-exponentially convex on J if it is n-exponentially convex in the Jensen sense and continuous on J. Remark 3.1. It is clear from the definition that 1-exponentially convex functions in the Jensen sense are nonnegative functions. Also, n-exponentially convex functions in the Jensen sense are k-exponentially convex in the Jensen sense for every k ∈ N, n ≥ k.

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The following proposition is a consequence of the well known Sylvester’s criterion. Proposition 3.2. If ψ is n-exponentially convex in the Jensen sense on J then the matrix n pi + pj ψ 2 i,j=1 is positive semi-definite. Particularly k pi + pj det ψ ≥ 0, 2 i,j=1 for 1 ≤ k ≤ n, and pi ∈ J, i = 1, . . . , k. Corollary 3.3. (i) If ψ : J → (0, ∞) is 2-exponentially convex in the Jensen sense then ψ is a log-convex function in the Jensen sense on J. (ii) If ψ : J → (0, ∞) is 2-exponentially convex then ψ is a log-convex function on I. Proof.

(i) From ξ12 ψ(x)

+ 2ξ1 ξ2 ψ

x+y 2

+ ξ22 ψ(y) ≥ 0,

for any ξ1 , ξ2 ∈ R and all x, y ∈ J, we conclude x+y ψ2 ≤ ψ(x)ψ(y), 2 for all x, y ∈ J. (ii) It follows from (i) using the continuity property.

(24)

Definition 2. A function ψ : J → R is exponentially convex in the Jensen sense on J if it is n-exponentially convex in the Jensen sense on J for every n ∈ N. A function ψ : J → R is exponentially convex if it is exponentially convex in the Jensen sense and continuous. Proposition 3.4. Let E denote the set of all exponentially convex functions on an open interval J. (i) (ii)

E is a convex cone i.e. if ψ, φ ∈ E and α, β ≥ 0 then αψ + βφ ∈ E. E is closed under multiplication i.e. if ψ, φ ∈ E then ψφ ∈ E.

Proof. Part (i) follows directly from the definition. Part (ii) is a consequence of the next theorem (see [4]). One of the main aspects of exponentially convex functions is its integral representation.

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Theorem 3.5. The function ψ : J → R is exponentially convex on J if and only if ∞ etx dσ(t), x ∈ J

ψ(x) =

(25)

−∞

for some non-decreasing function σ : R → R.

Proof. See [1, p. 211].

Remark 3.6. Every exponentially convex function is n-exponentially convex by 3 definition. The converse is not generally true, since for example ψ(x) = ex −x is 2-exponentially convex on (0, 1) and not an exponentially convex function on (0, 1) (see [4] for details). Theorem 3.7. Let Ω = {ψp : I → R, p ∈ J}, be a family of functions, such that p → [t0 , t1 , t2 ; ψp ] is n-exponentially convex in the Jensen sense on J for every three distinct points t0 , t1 , t2 ∈ I. If Wi , i = 1, 2, 3 are linear functionals defined as in (19) and (20), then the function p → Wi (ψp ) is n-exponentially convex in the Jensen sense on J. If the function p → Wi (ψp ) is continuous on J, then it is n -exponentially convex on J. Proof. We give the proof for the case i = 1. In the cases i = 2, 3 the proof is similar. For ξi ∈ R, i = 1, . . . , n and pi ∈ J, i = 1, . . . , n, we define the function n g(t) = ξi ξj ψ pi +pj (t). i,j=1

2

Using the assumption that the function p → [t0 , t1 , t2 ; ψp ] is n-exponentially convex in the Jensen sense, we have [t0 , t1 , t2 ; g] =

n

ξi ξj [t0 , t1 , t2 ; ψ pi +pj ] ≥ 0, 2

i,j=1

which in turn implies that g is a convex function on I and therefore from inequality (15) we have W1 (g) =

n i,j=1

ξi ξj W1 (ψ pi +pj ) ≥ 0. 2

Hence we conclude that the function p → W1 (ψp ) is n-exponentially convex on J in the Jensen sense. If the function p → W1 (ψp ) is also continuous on J, then p → W1 (ψp ) is n-exponentially convex by definition. The following corollary is the consequence of the above theorem.

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Corollary 3.8. Let Ω = {ψp : I → R, p ∈ J} be a family of functions, such that p → [t0 , t1 , t2 ; ψp ] is exponentially convex in the Jensen sense on J for every three distinct points t0 , t1 , t2 ∈ I. If Wi , i = 1, 2, 3 are linear functionals defined as in (19), (20) and (21), then the function p → Wi (ψp ) is exponentially convex in the Jensen sense on J. If the function p → Wi (ψp ) is continuous on J, then it is exponentially convex on J. Corollary 3.9. Let Ω = {ψp : I → R, p ∈ J}, be a family of functions, such that p → [t0 , t1 , t2 ; ψp ] is 2-exponentially convex on J for every three distinct points t0 , t1 , t2 ∈ I. If Wi , i = 1, 2, 3 are linear functionals defined as in (19), (20) and (21), then the following statements hold: (i) If the function p → Wi (ψp ) is continuous, it is a 2-exponentially convex function on J, thus log-convex on J and for p, q, r ∈ I such that p < q < r we have, r−q

(Wi (ψp )) (ii)

(Wi (ψr ))

q−p

r−p

≥ (Wi (ψq ))

.

(26)

If the function p → Wi (ψp ) is strictly positive and differentiable on J, then for every p, q, r, s ∈ J such that p ≥ r, q ≥ s, Ep,q (Wi ; Ω) ≥ Er,s (Wi ; Ω),

(27)

holds, where

⎧

1 Wi (ψp ) q−p ⎪ ⎪ , ⎨ Wi (ψq ) d Ep,q (Wi ; Ω) = ⎪ (Wi (ψp )) ⎪ ⎩ exp dpW (ψ , i p)

p, q ∈ J, p = q, (28) q = p ∈ J,

ψp , ψq ∈ Ω. Proof. We prove the statements for i = 1 and the proofs for the cases i = 2, 3 are similar. (i) This is an immediate consequence of Theorem 3.7 and Corollary 3.3 (ii). (ii) Since by (i) the function p → W1 (ψp ) is log-convex on J, that is, the function p → log W1 (ψ p ) is convex on J, we get log W1 (ψp ) − log W1 (ψq ) log W1 (ψr ) − log W1 (ψs ) ≥ s−q r−s

(29)

for p ≥ r, q ≥ s, p = q, r = s, and so we conclude that Ep,q (W1 ; Ω) ≥ Er,s (W1 ; Ω). It is easy to see that limq→p Ep,q (W1 ; Ω) = Ep,p (W1 ; Ω). For the case p = q, consider limq→p in (29) and conclude that Ep,p (W1 ; Ω) ≥ Er,s (W1 ; Ω). The case r = s can be treated similarly.

(30)

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Remark 3.10. Note that the results from Theorem 3.7, Corollary 3.8 and Corollary 3.9 still hold when two of the points t0 , t1 , t2 ∈ I coincide for a family of differentiable functions ψp such that p → [t0 , t1 , t2 ; ψp ] is n-exponentially convex in the Jensen’sense (exponentially convex in the Jensen’sense), further, they still hold when all three points coincide for a family of twice differentiable functions with the same property. The proofs are obtained using Theorem 3.7, Corollaries 3.8, 3.9 and C2 and C3 .

4. Examples and applications Example 4.1. Consider a family of functions Ω1 = {ψp : R → [0, ∞), p ∈ R}, defined with

ψp (t) =

ept p2 , 1 2 2t ,

p = 0, p = 0.

2

d pt Since dt and p → ept is obviously an exponentially convex func2 (ψp (t)) = e tion on R for each t ∈ R, it is easy to prove that the function p → [t0 , t1 , t2 ; ψp ] is also exponentially convex for arbitrary real numbers t0 , t1 , t2 . Using Corollary 3.8, it follows that p → Wk (ψp ) is exponentially convex (it is easy to verify that it is continuous) and thus log-convex. It follows that the expression (28) becomes 1 W1 (ψp ) p−q , p = q, Ep,q (W1 ; Ω1 ) = W1 (ψq ) ⎞ ⎛ (b+x)ep(b+x) −(a−x)ep(a−x) bepb −aepa − −3 2x+b−a b−a ⎠ , p = 0, + Ep,p (W1 ; Ω1 ) = exp ⎝ ep(b+x) −ep(a−x) epb −epa p − 2x+b−a b−a a+b E0,0 (W1 ; Ω1 ) = exp . 2

Assuming that f is a nonnegative concave function on [a, b] and using Corollary 3.8, it follows that p → Wk (ψp ) , k = 2, 3 is exponentially convex and thus log-convex. In this case (28) becomes

1 W2 (ψp ) p−q , p = q, W2 (ψq ) ⎛ 3−pδ epδ+ − 3−pδ epδ− ⎞ b ( ( +) −) 1 (pf (t) − 2) epf (t) dt + b−a a p δ −δ ( + −) ⎜ ⎟ Ep,q (W2 ; Ω1 ) = exp ⎝ ⎠ , p = 0, b pδ pδ p pf (t) dt − e + −e − e b−a a δ −δ

Ep,q (W2 ; Ω1 ) =

+

−

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⎞ 4 4 δ+ −δ− − M33 (f ) ⎟ ⎜ 1 4(δ+ −δ− ) E0,0 (W2 ; Ω1 ) = exp ⎝ ⎠ 3 3 3 δ+ −δ− − M 2 (f ) 2 3(δ+ −δ− ) ⎛

where δ+ = 2f − fmin and δ− = fmin . All possible cases for log Ep,q (W3 ; Ω1 ) can be obtained by replacing δ− by d− (d− = 2f − fmax ) and δ+ by d+ (fmax ) in log Ep,q (W2 ; Ω1 ) respectively. Note that, Ep,q (Wk ; Ω1 ), k = 1, 2, 3, satisfy the monotonicity property, that is for p ≥ r and q ≥ s Ep,q (Wk ; Ω1 ) ≥ Er,s (Wk ; Ω1 ) holds. Observe here, that according to Theorem 2.7, we have Cauchy means: d ≥ Mp,q (a, b) = log Ep,q (W1 ; Ω1 ) ≥ 0 and also for k = 2, 3: fmax ≥ Mp,q (f ) = log Ep,q (Wk ; Ω1 ) ≥ fmin . Example 4.2. Consider a family of functions Ω2 = {gp : (0, ∞) → (0, ∞), p > 0}, defined by

√

e−t gp (t) = p d2 g (t)

p

.

√

Since p → dtp2 = e−t p , being the Laplace transform of a non-negative function, is exponentially convex on (0, ∞) (see [13]). Clearly, gp is a convex function for each p > 0. It is easy to prove that the function p → [t0 , t1 , t2 ; gp ] is also exponentially convex for arbitrary positive t0 , t1 , t2 . It is obvious that p → Wk (gp ) is continuous for k = 1, 2, 3. Using Corollary 3.8 and assuming that f is non-negative concave on [a, b] (only for W2 (gp ) and W3 (gp ) ), it follows that p → Wk (gp ) is exponentially convex and thus log-convex. For this family of functions, (28) becomes ⎧

⎨ 1 log W1 (gp ) , p = q, p−q W1 (gq ) log Ep,q (Wk ; Ω2 ) = −1 ⎩ √ log E−√p,−√p (Wk ; Ω1 ), q = p. p Note that Ep,q (Wk ; Ω2 ), k = 1, 2, 3 satisfy the monotonicity property, that is for p ≥ r and q ≥ s Ep,q (Wk ; Ω2 ) ≥ Er,s (Wk ; Ω2 ) holds. Observe that according to Theorem 2.7, we have √ √ d ≥ Mp,q (a, b) = −( p + q) log Ep,q (W1 ; Ω2 ) ≥ 0 and also for k = 2, 3:

√ √ fmax ≥ Mp,q (f ) = −( p + q) log Ep,q (Wk ; Ω2 ) ≥ fmin .

150

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So Mp,q (a, b) and Mp,q (f ) can be regarded as Cauchy means. Example 4.3. Consider a family of functions Ω3 = {hp : (0, ∞) → (0, ∞), p > 0}, defined with

⎧ ⎨ hp (t) =

p−t (log p)2 ,

p = 1,

⎩t , 2 2

p = 1.

2

d −t is exponentially convex by Example 4.1. Obviously hp p → dt 2 hp (t) = p is a convex function for every p > 0. It is easy to prove that the function p → [t0 , t1 , t2 ; hp ] is also exponentially convex for arbitrary positive real numbers t0 , t1 , t2 . Assuming that f is a non-negative concave function on [a, b] (only for W2 (hp ) and W3 (hp ) ) and using Corollary 3.7, the exponential convexity of p → Wk (gp ) is obtained (it is easy to verify that it is continuous) and thus also its log-convexity. For this particular family of functions, the expression (28) becomes:

⎧ Wk (hp ) 1 ⎪ , p = q, log ⎪ p−q W ⎨ k(h )

log Ep,q (Wk ; Ω3 ) =

⎪ ⎪ ⎩

−1 p

q

log (E− log p,− log p (Wk ; Ω1 )),

− log (E0,0 (Wk ; Ω1 )),

q = p = 1,

q = p = 1.

Note that Ep,q (Wk ; Ω3 ), k = 1, 2, 3 satisfies the monotonicity property, that is for p ≥ r and q ≥ s Ep,q (Wk ; Ω3 ) ≥ Er,s (Wk ; Ω3 ) holds. Observe that according to Theorem 2.7, we have Cauchy means: d ≥ Mp,q (a, b) = −L(p, q) log Ep,q (W1 ; Ω3 ) ≥ 0 and also for k = 2, 3: fmax ≥ Mp,q (f ) = −L(p, q) log Ep,q (Wk ; Ω3 ) ≥ fmin where L(p, q) is the logarithmic mean defined by L(p, q) = q, L(p, p) = p.

p−q log p−log q , p

=

Example 4.4. Consider a family of functions Ω4 = {φp : (0, ∞) → R; p ∈ R} defined by

⎧ tp ⎨ p(p−1), φp (t) = − ln t, ⎩ t ln t,

p = 1, 0, p = 0, p = 1.

(31)

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2

d p−2 Since p → dt = e(p−2) ln t is the Laplace transform of a non-neg2 (φp (t)) = t ative function (see [13]), it is exponentially convex. Obviously φp is a convex function for every p > 0. In the same manner as in the previous examples, it follows that p → W1 (φp ) is exponentially convex (it is easy to verify that it is continuous), and thus log-convex. It follows from (28) that

1 W1 (φp ) p−q , p = q, W1 (φq ) ⎞ ⎛ p+1 p+1 (b+x)p+1 ln(b+x)−(a−x)p+1 ln(a−x) ln a − b ln b−a 1 − 3p2 2x+b−a b−a ⎠, Ep,p (W1 ; Ω4 ) = exp ⎝ 3 + p+1 p+1 (b+x)p+1 −(a−x)p+1 p −p − b −a

Ep,q (W1 ; Ω4 ) =

(2x+b−a)

b−a

p = −1, 0, 1, ⎞ ⎛ 2 2 ln2 (b+x)−ln2 (a−x) a − ln b−ln 3 1 2x+b−a b−a ⎠, E−1,−1 (W1 ; Ω4 ) = exp ⎝ + b−ln a 2 2 ln(b+x)−ln(a−x) − ln b−a 2x+b−a ⎛ ⎞ 2 (b+x) ln2 (b+x)−(a−x) ln2 (a−x) ln2 a − b ln b−a 1 2x+b−a b−a ⎠, E0,0 (W1 ; Ω4 ) = exp ⎝ ln(a−x) ln a 2 (b+x) ln(b+x)−(a−x) − b ln b−a 2x+b−a b−a ⎞ ⎛ 2 2 2 (b+x)2 ln2 (b+x)−(a−x)2 ln2 (a−x) ln2 a − b ln b−a 3 1 2x+b−a b−a ⎠. ⎝ E1,1 (W1 ; Ω4 ) = exp − + 2 2 2 2 (b+x) ln(b+x)−(a−x) ln(a−x) − b2 ln b−a2 ln a 2x+b−a b−a

Assuming that f is a nonnegative concave function on [a, b] and using Corollary 3.8, it follows that p → Wk (φp ), k = 2, 3 is exponentially convex and thus log-convex. Now (28) becomes

Ep,q (W2 ; Ω4 ) =

W2 (ψp ) W2 (ψq ) ⎛

1 p−q

p = q,

,

⎜ 1 − 2p Ep,p (W2 ; Ω4 ) = exp ⎝ − p(p − 1)

⎜3 E−1,−1 (W2 ; Ω4 ) = exp ⎝ + 2 E0,0 (W2 ; Ω4 ) = exp ⎝

(1+p)2 (δ+ −δ− ) p+1 p+1 δ+ −δ− (1+p)(δ+ −δ− )

p = −1, 0, 1,

⎛

⎛

p+1 (1−(1+p) ln δ+ )δ+p+1 −(1−(1+p) ln δ− )δ−

ln2 δ+ −ln2 δ− 2(δ+ −δ− ) ln δ+ −ln δ− δ+ −δ−

− Π−1 (f )

− M−1 (f )

− Mpp (f )

⎞ + Πp (f ) ⎟ ⎠,

⎞ ⎟ ⎠,

⎞ − 2M0 (f ) − Π0 (f ) ⎠, δ ln δ −δ ln δ− 2 + δ++ −δ− − 1 − M0 (f ) −

δ+ ln2 δ+ −δ− ln2 δ− δ+ −δ−

E1,1 (W2 ; Ω4 ) ⎛ 2 2 ⎞ 2 b δ+ (ln δ+ −3 ln δ+ )−δ− (ln2 δ− −3 ln δ− ) 3M1 (f ) 1 2 + − f (t) ln f (t) + 2Π (f ) 1 ⎜ ⎟ 2 b−a a 2(δ+ −δ− ) ⎟. = exp⎜ 2 ln δ −δ 2 ln δ ⎝ ⎠ δ+ + + − + M1 (f ) − 2Π1 (f ) δ −δ +

where,

⎧ ⎨ Πp (f ) =

⎩

1 b−a 1 b−a

−

b a

b a

f p (t) ln f (t)dt, 2

ln f (t)dt,

p = 0, p=0

152

and

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⎧ ⎪ ⎨

p1 p f (t)dt , a Mp (f ) =

⎪ ⎩ exp 1 b ln f (t)dt , b−a a 1 b−a

b

p = 0, p = 0.

All possible cases for Ep,q (W3 ; Ω4 ) can be obtained by replacing δ− by d− and δ+ by d+ respectively. Note that, for this family of functions, Theorem 2.7 implies 1 W1 (φp ) p−q d≥ ≥0 W1 (φq ) and

fmax ≥

Wk (φp ) Wk (φq )

1 p−q

≥ fmin ,

k = 2, 3.

So, Ep,q (Wk ; Ω4 ), k = 1, 2, 3, can be regarded as Cauchy means. Also observe that Ep,q (Wk ; Ω4 ) are monotonic means, that is for p ≥ r and q ≥ s, Ep,q (Wk ; Ω4 ) ≥ Er,s (Wk ; Ω4 ). Remark 4.5. Some related results as consequences of (3) and (2) were obtained in [11] and [10]. Corollary 4.6. Suppose that f is non-negative concave on [a, b] (a, b ∈ R) and ψ is continuous convex on [fmin , fmax ]. If W2 (ψ) is as defined in (20) and p ∈ R\{−1, 0, 1}, q, r ∈ R with q < p < r, then the following estimation is obtained: ! p+1 p+1 r−p p−q − δ− δ+ 1 p − Mp (f ) . (32) [W2 (φq )] r−q [W2 (φr )] r−q ≥ p(p − 1) (p + 1)(δ+ − δ− ) If p < r < q or r < q < p, then the inequality sign is reversed. Proof. By setting p = p, q = p for k = 2 in (26), we obtain r−p

p−q

[W2 (ψq )] r−q [W2 (ψr )] r−q ≥ [W2 (ψp )].

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Remark 4.7. If a concave function f is such that fmin = 0 and fmax = 1, then p+1 p+1 δ+ − δ− 2p M1p (f ) − Mpp (f ) = − Mpp (f ). (p + 1)(δ+ − δ− ) 1+p

Obviously, (32) is an improvement of the second inequality in [14, Corollary 4.2], since (32) gives an upper bound and the converse of (32) gives a positive lower bound for the difference given by the second inequality in [14, Corollary

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4.2]. Note also that [14, Corollary 4.2] holds only for p ≥ 1, but the estimation (32) is valid for any p ∈ R. The cases p = −1, 0, 1 can be covered simultaneously from (33) choosing appropriate ψp . Corollary 4.8. Suppose that f is non-negative concave on [a, b] (a, b ∈ R) and ψ is continuous convex on [fmin , fmax ]. If W3 (ψ) is as defined in (20) and p, q, r ∈ R (p = −1, 0, 1) with q < p < r, then the following estimation is obtained: ! r−p p−q − dp+1 dp+1 1 + − p r−q r−q [W3 (φq )] Mp (f ) − . (34) [W3 (φr )] ≥ p(p − 1) (p + 1)(d+ − d− ) If p < r < q or r < q < p, then inequality sign is reversed. Proof. By setting p = p, q = p for k = 3 in (26), we obtain r−p

p−q

[W3 (ψq )] r−q [W3 (ψr )] r−q ≥ [W3 (ψp )].

(35)

Remark 4.9. If a concave function f is such that fmin = 0 and fmax = 1, then Mpp (f ) −

p+1 − dp+1 dp+1 −1 (2M1 (f ) − 1) + − = Mpp (f ) − . (p + 1)(d+ − d− ) 2(p + 1)(M1 (f )−)

Obviously, (34) is an improvement of the first inequality in [14, Corollary 4.2], since (34) gives an upper bound and the converse of (34) gives a positive lower bound for the difference given by the first inequality in [14, Corollary 4.2]. Note also that [14, Corollary 4.2] holds only for p ≥ 1, but the estimation (34) is valid for p ∈ R. The cases p = −1, 0, 1 can be covered simultaneously from (35) choosing appropriate ψp .

References [1] Akhiezer, N.I.: The Classical Moment Problem and Some Related Questions in Analysis. Oliver and Boyd, Edinburgh (1965) [2] Bernstein, S.N.: Sur les fonctions absolument monotones. Acta Math. 52, 1–66 (1929) ˇ [3] Culjak, V., Franji´ c, I., Peˇ cari´ c, J., Roqia, G.: Schur-convexity of averges of convex functions. J. Inequal. Appl. 2011 (Art. ID 581918) [4] Jakˇseti´ c, J., Peˇ cari´ c, J.: Exponential Convexity Method. J. Convex Anal. (2013, to appear) [5] Jakˇseti´ c, J., Peˇ cari´ c, J., Roqia, G.: On Jensen’s inequality involving averages of convex functions. Sarajevo J. Math. 8(20) (2012) [6] Karlin, S., Studden, W.J.: Tchebycheff Systems: With Applications in Analysis and Statistics. Interscience, New York (1966) [7] Peˇ cari´ c, J., Peri´ c, I., Roqia, G.: Exponentially convex functions generated by Wulberts inequality and Stolarsky-type means. Math. Comput. Model. 55, 1849–1857 (2012) [8] Peˇ cari´ c, J., Peri´ c, J.: Improvements of the Giaccardi and the Petrović inequality and related Stolarsky type means. Ann. Univ. Craiova Ser. Mat. Inform. 39(1):65–75 (2012)

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[9] Peˇ cari´ c, J.E., Proschan, F., Tong, Y.L.: Convex functions, partial orderings, and statistical applications. Academic Press, Inc., London (1992) [10] Latif, N., Peˇ cari´ c, J., Peri´ c, I.: Exponential convexity of the Favard’s inequality and related results. Makedon. Akad. Nauk. Umet. Oddel. Mat.-Tehn. Nauk. Prilozi. 30(1– 2), 53–66 (2010) [11] Latif, N., Peˇ cari´ c, J., Peri´ c,I.: Some new results related to the Favard’s inequality. J. Inequal. Appl. 2009 (Art. ID 128486) [12] Roberts, A.W., Varberg, D.E.: Convex Functions. Academic Press, New York (1973) [13] Widder, D.V.: The Laplace Transform. Princeton University Press, New Jersey (1941) [14] Wulbert, D.E.: Favard’s inequality on average values of convex functions. Math. Comput. Model. 37, 1383–1391 (2003)

J. Peˇ cari´ c Faculty of Textile Technology University of Zagreb Pierottijeva 6 10000 Zagreb, Croatia e-mail: [email protected] I. Peri´ c Faculty of Food Technology and Biotechnology University of Zagreb 10000 Zagreb, Crotia e-mail: [email protected] G. Roqia and I. Peri´ c Abdus Salam School of Mathematical Sciences 68-B, New Muslim Town Lahore 54000, Pakistan e-mail: [email protected] Received: June 21, 2012 Revised: October 14, 2012