INTEGRAL INEQUALITIES FOR QUASICONVEX

INTEGRAL INEQUALITIES FOR QUASICONVEX FUNCTIONS AND APPLICATIONS ;F

ÇET I·N YILDIZ

AND MUSTAFA GÜRBÜZ H

Abstract. In this paper, using integral representations for n times di¤erentiable mappings, we establish new generalizations of certain Hermite-Hadamard type inequality for quasiconvex functions. We also give some new estimations for special means of real numbers.

1. INRODUCTION Let f : I R ! R be a convex function de…ned on an interval I of real numbers, a; b 2 I and a < b: The following double inequality is well known in the literature as Hadamard’s inequality: Z b a+b 1 f (a) + f (b) (1.1) f : f (x)dx 2 b a a 2

Both inequalities hold in the reversed direction if f is concave. The inequalities in (1.1) have become an important cornerstone in mathematical anlysis and optimization. Many uses of these inequalities have been discovered in a variety of settings. Moreover, many inequalities of special means can be obtained for a particular choice of the function f . Due to the rich geometrical signi…cance of Hermite-Hadamard inequlity, there is growing literature providing its new proofs, extensions, re…nements and generalizations, see for example ( [1], [4], [5], [9], [11], [14], [16]-[19]) and the references therein.

De…nition 1. A function f : [a; b] R ! R is said to be convex if whenever x; y 2 [a; b] and t 2 [0; 1], the following inequality holds: f (tx + (1

t)y)

tf (x) + (1

t)f (y):

We say that f is concave if ( f ) is convex. This de…nition has its origins in Jensen’s results from [10] and has opened up the most extended, useful and multidisciplinary domain of mathematics, namely, convex analysis. Convex curves and convex bodies have appeared in mathematical literature since antiquity and there are many important results related to them. We recall that the notion of quasiconvex functions generalizes the notion of convex functions. De…nition 2. A function f : [a; b] f (tx + (1

R ! R is said to be quasiconvex on [a; b] if

t)y)

max ff (x); f (y)g ;

2000 Mathematics Subject Classi…cation. 26D15, 26A51. Key words and phrases. Hermite-Hadamard Integral Inequality, Hölder Inequality, Jensen Inequality, Convex Functions. F Corresponding Author. 1

ÇET I·N YILDIZ

2

AND M USTAFA GÜRBÜZ H

;F

for all x, y 2 [a; b] and t 2 [0; 1]: Clearly, any convex function is quasiconvex. Furthermore, there exist quasiconvex functions which are not convex (see [5], [9], [14], [21], [22]). For example, consider the following: Let f : R+ ! R; f (x) = ln x; x 2 R+ : This function is quasiconvex. However f is not a convex function. Y¬ld¬z (see [20]) proved the following generalization for n-time di¤erentiable functions. Lemma 1. For n f (n) 2 L[a; b], then (1.2) Z b

f (t)dt

=

a

n X1 k=0

1; let f : [a; b] ! R be n-time di¤ erentiable functions. If

1 + ( 1)k 2k+1 (k + 1)!

(b

(b a)n+1 2n+1 n!

Z

+( 1)n

1

tn f (n) t

0

+

a+b 2

a)k+1 f (k)

Z

a+b + (1 2

t)a dt

1

1)n f (n) tb + (1

(t

t)

0

a+b 2

dt :

For other recent results concerning the n times di¤erentiable functions and quasiconvex functions see [1]-[7], [6]-[8], [11]-[13], [18] where further references are given. The main purpose of the this paper is to establish several new inequalities for n time di¤erantiable mappings that are connected with the celebrated HermiteHadamard integral inequality for quasiconvex functions.

2. MAIN RESULTS Theorem 1. Let f : [a; b] R ! R be n time di¤ erentiable function and a < b: q If f (n) 2 L[a; b] and f (n) (n 1) is quasiconvex on [a; b]; then we have: Z

(2.1)

b

f (x)dx

a

(b a)n+1 2n+1 n!

n X1 k=0

1 np + 1

1 + ( 1)k + 1)! " 1

2k+1 (k p

max

+ max

where

1 p

+

1 q

= 1:

(b

a+b 2

a)k+1 f (k) q

a+b 2

f (n) (a) ; f (n)

f

(n)

a+b 2

q

; f

(n)

q

q

(b)

1 q

1 q

#

3

Proof. Using Lemma 1 and Hölder integral inequality, it follows that Z b n X1 1 + ( 1)k a+b f (x)dx (b a)k+1 f (k) k+1 (k + 1)! 2 2 a k=0 ( Z 1 Z 1 q 1 p (b a)n+1 a+b (n) np f t dt t dt + (1 t)a 2n+1 n! 2 0 0 +

Z

Z

1 p

1 np

(1

t) dt

0

1

f

(n)

1 q

tb + (1

0

1 q

q

a+b t) 2

dt

)

:

q

Since f (n) is quasiconvex on [a; b]; then we can write Z b n X1 1 + ( 1)k a+b f (x)dx (b a)k+1 f (k) k+1 2 (k + 1)! 2 a k=0 ( 1 Z 1 p q (b a)n+1 a+b 1 max f (n) (a) ; f (n) n+1 2 n! np + 1 2 0 +

=

(b a)n+1 2n+1 n!

1 p

1 np + 1

1 np + 1

1 p

"

Z

1

f

max

(n)

0

f

max

(n)

q

(a) ; f

f (n)

+ max

; f

(n)

q

a+b 2

; f (n) (b)

(b)

dt 1 q

1 q

q

which completes the proof. Corollary 1. Under the assumptions of Theorem 1, we have Z b n X1 1 + ( 1)k f (x)dx (2.2) (b a)k+1 f (k) k+1 2 (k + 1)! a k=0 " q q a+b (b a)n+1 (n) (n) f ; max f (a) 2n+1 n! 2 + max

f

a+b 2

(n)

q

; f

(n)

a+b 2 1 q

q

1 q

(b)

Proof. For p > 1; since lim

p!1

1 np + 1

1 p

= 1 and

we have 1 1 < lim p!1 n+1 np + 1 Hence we obtain the inequality (2.2).

lim+

p!1

1 np + 1

1 p

=

1 ; n+1

1 p

< 1;

Corollary 2. Let f as in Theorem 1, if in addition

p 2 (1; 1):

1 q

q

q

a+b 2

(n)

dt

q

a+b 2

1 q

q

#

#

)

;F

ÇET I·N YILDIZ

4

AND M USTAFA GÜRBÜZ H

(1) f (n) is increasing, then we obtain Z

n X1

b

f (x)dx

a

1 + ( 1)k 2k+1 (k + 1)!

k=0

n+1

(b a) 2n+1 n!

1 p

1 np + 1

a+b 2

f (n)

a+b 2

a)k+1 f (k)

(b

+ f (n) (b)

:

(2) f (n) is decreasing, then we obtain Z

n X1

b

f (x)dx

a

1 + ( 1)k 2k+1 (k + 1)!

k=0

n+1

(b a) 2n+1 n!

1 p

1 np + 1

a+b 2

a)k+1 f (k)

(b

a+b 2

f (n) (a) + f (n)

:

Theorem 2. Let f : [a; b] R ! R be n time di¤ erentiable function and a < b: q If f (n) 2 L[a; b] and f (n) (n 1) is convex on [a; b]; then we get Z

(2.3)

n X1

b

f (x)dx

a

1 + ( 1)k 2k+1 (k + 1)!

k=0

n+1

(b a) 2n+1 n! "

1

q 1 nq + q p q

f

+ max

(n)

a+b 2

q

a+b 2

1 q

1 p+1

1

f (n) (a) ; f (n)

max

1 q

; f

a+b 2

a)k+1 f (k)

(b

(n)

q

1 q

q

1 q

(b)

#

where p > 1: Proof. From Lemma 1 and using the properties of modulus, we get Z

b

f (x)dx

a n+1

(b a) 2n+1 n!

=

(b a)n+1 2n+1 n!

n X1

2k+1 (k

t

f (n) t

1 + ( 1)k + 1)!

k=0 1 n

Z

0

Z

+ (Z

t)a

t)n f (n) tb + (1

0 p

1 n

t tq t

0

+

a+b + (1 2

Z

0

dt

1

(1

1

p q

(1

f (n) t

(1

a+b + (1 2

t)

a+b 2

t)a

p

t)n (1

t) q p

t) q

f

a+b 2

a)k+1 f (k)

(b

(n)

tb + (1

dt

dt a+b t) 2

)

dt :

5

Using the Hölder integral inequality, we can write Z

n X1

b

a+b 1 + ( 1)k (b a)k+1 f (k) + 1)! 2 a k=0 8 1 Z 1 n q q 1 !1 q1 Z 1 q q t a+b (b a)n+1 < (n) p t dt t f dt + (1 t)a p 2n+1 n! : 0 t q 2 0 11 q1 0 # q Z 1 Z 1" n q 1 a+b (1 t) A @ dt (1 t)p f (n) tb + (1 t) + p q 2 (1 t) 0 0 Since f (n)

f (x)dx

q

2k+1 (k

is quasiconvex on [a; b]; we have Z

n X1

b

f (x)dx

a

1 + ( 1)k + 1)!

k=0

n+1

(b a) 2n+1 n! "

f

max

+ max

(n)

f

(a) ; f

q

a+b 2

; f

(n)

1 q

q

a+b 2

(n)

1 q

1 p+1

1

q

(n)

1 q

1

q 1 nq + q p

a+b 2

a)k+1 f (k)

(b

2k+1 (k

1 q

q

(b)

#

which completes the proof of the theorem.

Corollary 3. In Theorem 2, if we choose n = 1, we have

1

a+b 2

f (b

b

a) 4 "

q

a

b

f (x)dx

a 1

1

2q

Z

p q

0

f

0

0

a+b 2

1 q

1 p+1

1

max jf (a)j ; f

+ max

1 q

1 q

q

a+b 2 q 0

q

; jf (b)j

1 q

#

:

1 q

q

dt

9 > = > ;

:

ÇET I·N YILDIZ

6

AND M USTAFA GÜRBÜZ H

;F

Corollary 4. In Theorem 2, if we choose n = 2, then we obtain Z b a+b 1 f f (x)dx 2 b a a (b

2

a) 16 "

q

1

1

3q

p

max jf (a)j ; f f

+ max

00

a+b 2 00

; jf (b)j

k=0

(b a) 2n+1 n!

1

q 1 nq + q p

1 q

1

1 p+1

(2) f (n) is decreasing, then we obtain Z b n X1 1 + ( 1)k f (x)dx (b k+1 2 (k + 1)! a

1 q

f (n)

a)k+1 f (k)

k=0

n+1

(b a) 2n+1 n!

q 1 nq + q p

1

1

1 q

1 q

q

Corollary 5. Let f as in Theorem 2, if in addition (1) f (n) is increasing, then we obtain Z b n X1 1 + ( 1)k (b a)k+1 f (k) f (x)dx 2k+1 (k + 1)! a n+1

1 q

q

q

a+b 2

00

1 q

1 p+1

1 q

00

1 q

1 p+1

1 q

#

:

a+b 2 a+b 2

+ f (n) (b)

:

a+b 2

:

a+b 2

f (n) (a) + f (n)

Theorem 3. For n 1; let f : [a; b] ! R be n time di¤ erentiable function and q a < b: If f (n) 2 L[a; b] and f (n) is convex on [a; b]; for q 1; then the following inequality holds: Z b n X1 1 + ( 1)k a+b (2.4) f (x)dx (b a)k+1 f (k) k+1 2 (k + 1)! 2 a k=0 " 1 q q q (b a)n+1 a+b (n) (n) max f (a) ; f 2n+1 (n + 1)! 2 1# q q q a+b (n) (n) + max f ; f (b) : 2 Proof. Suppose that q = 1: From Lemma 1, we have Z b n X1 1 + ( 1)k (b a)k+1 f (k) f (x)dx 2k+1 (k + 1)! a k=0

(b a)n+1 max 2n+1 (n + 1)!

+ max

f (n) (a) ; f (n) f (n)

a+b 2

a+b 2

a+b 2

; f (n) (b)

:

7

Suppose now that q > 1. Using the well known Power-mean integral inequality and Lemma 1, we obtain Z b n X1 1 + ( 1)k a+b f (x)dx (b a)k+1 f (k) k+1 2 (k + 1)! 2 a k=0 ( Z 1 1 Z 1 q q 1 1 q (b a)n+1 a+b n n (n) dt t dt t f + (1 t)a t 2n+1 n! 2 0 0 Z

+

Z

1 q

1

1

t)n dt

(1

0

Since f (n) Z

q

n X1

b

f (x)dx

k=0

(b a)n+1 2n+1 n!

1 + ( 1)k + 1)! ( 1 Z

1

t

max

f

(n)

q

(a) ; f

Z

1

(1

n

t)

max

f

a+b 2

(n)

0

max

+ max

(n)

0

+ (

n

f

(n)

q

(a) ; f

f (n)

1 q

q

dt

a+b 2

(n)

a+b 2

q

; f (n) (b)

q

1 q

q

1 q

)

dt

q

; f

(n)

(b)

:

(1) f (n) is increasing, then we obtain f (x)dx

a

n X1 k=0

(b a)n+1 2n+1 (n + 1)!

f (n)

1 + ( 1)k 2k+1 (k + 1)! a+b 2

(b

a)k+1 f (k)

+ f (n) (b)

a+b 2

:

(2) f (n) is decreasing, then we obtain Z

b

f (x)dx

a

n X1 k=0

n+1

(b a) 2n+1 (n + 1)!

1 + ( 1)k + 1)!

2k+1 (k

f (n) (a) + f (n)

(b a+b 2

a)k+1 f (k)

a+b 2

:

3. APPLICATIONS TO SPECIAL MEANS We now consider the means for arbitrary real numbers ;

1 q

q

Corollary 6. Let f as in Theorem 3, if in addition

b

:

1 q

q

a+b 2

Hence, the proof of the theorem is completed.

Z

)

a+b 2

a)k+1 f (k)

1

q

a+b t) 2

1; then we obtain

(b

2k+1 (k

1 n+1

(b a)n+1 2n+1 (n + 1)!

t)n f (n) tb + (1

(1

0

is quasiconvex on [a; b]; for q

a

=

1

( 6= ): We take

dt

)

;F

ÇET I·N YILDIZ

8

AND M USTAFA GÜRBÜZ H

(1) Arithmetic mean : + ; 2

A( ; ) =

2 R+ :

;

(2) Logarithmic mean: L( ; ) =

ln j j

(3) Generalized log

ln j j

j j= 6 j j;

;

6= 0;

2 R+ :

;

mean: n+1

Ln ( ; ) =

;

1 n

n+1

(n + 1)(

;

)

n 2 Znf 1; 0g;

2 R+ :

;

Now using the results of Section 2, we give some applications for special means of real numbers. Proposition 1. Let a; b 2 R+ , 0 < a < b and n 2 N; n > 1: Then, we have n

jA (a; b)

Lnn (a; b)j

n

(b

q

a) 4

2

2q

1 q

1

1 p 1 (

qn q

4 max jaj + max

(

;

a+b 2

1 p+1

1 q

qn q

a+b 2 qn q

qn q

; jbj

)! q1 )! q1 3

5:

Proof. The assertion follows from Corollary 3 applied for f (x) = xn ; x 2 R; n 2 N: Proposition 2. Let a; b 2 R+ , a < b: Then, we have n

jA (a; b)

Lnn (a; b)j

(n2

n) (b 16

2

a)

1

max

(

1

1 q

1

q 1 3q p 1 p+1 2 ( )! q1 3q 3q 2 16 4 max ; a a+b

+

q

16 a+b

3q

2 ; b

3q

)! q1 3 5

Proof. The assertion follows from Corollary 4 applied for f (x) = x1 ; x 2 [a; b]: References

[1] S.-P. Bai, S.-H. Wang, F. Qi, Some Hermite-Hadamard type inequalities for n-time di¤erentiable ( ; m)-convex functions, J. Ineq. and Appl., 2012, 2012:267. [2] P. Cerone, S.S. Dragomir, J. Roumeliotis, Some Ostrowski type inequalities for n-time differentiable mappings and applications, Demonstratio Math., 32 (4) (1999), 697-712. [3] P. Cerone, S.S. Dragomir, J. Roumeliotis, J. Šunde, A new generalization of the trapezoid formula for n-time di¤erentiable mappings and applications, Demonstratio Math., 33 (4) (2000), 719-736.

9

[4] S.S. Dragomir, C.E.M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University, 2000. Online:[http://www.staxo.vu.edu.au/RGMIA/monographs/hermite hadamard.html]. [5] M. Alomari, M. Darus and U.S. K¬rmaci, Re…nements of Hadamard-type inequalities for quasi-convex functions with applications to trapezoidal formula and to special means, Comp. Math. Appl., 59 (2010), 225-232. [6] P. Cerone, S.S. Dragomir and J. Roumeliotis, Some Ostrowski type inequalities for n-time di¤erentiable mappings and applications, Demonstratio Math., 32 (4) (1999), 697-712. [7] P. Cerone, S.S. Dragomir and J. Roumeliotis and J. Šunde, A new generalization of the trapezoid formula for n-time di¤erentiable mappings and applications, Demonstratio Math., 33 (4) (2000), 719-736. [8] D.-Y. Hwang, Some Inequalities for n-time Di¤erentiable Mappings and Applications, Kyung. Math. Jour., 43 (2003), 335-343. [9] D.A. Ion, Some estimates on the Hermite-Hadamard inequalities through quasi-convex functions, Annals of University of Craiova, Math. Comp. Sci. Ser., 34 (2007), 82-87. [10] J. L. W. V. Jensen, On konvexe funktioner og uligheder mellem middlvaerdier, Nyt. Tidsskr. Math. B., 16, 49-69, 1905. [11] W.-D. Jiang, D.-W. Niu, Y. Hua, F. Qi, Generalizations of Hermite-Hadamard inequality to n-time di¤erentiable function which are s-convex in the second sense, Analysis (Munich), 32 (2012), 209-220 [12] M.E. Özdemir, Ç. Y¬ld¬z, New Inequalities for n-time di¤erentiable functions, Arxiv:1402.4959v1. [13] M.E. Özdemir, Ç. Y¬ld¬z, A New Generalization of the Midpoint Formula for n-Time Di¤erentiable Mappings which are Convex, RGMIA Research Report Collection, 17 (2014), Article 48. [14] M.E. Özdemir, Ç. Y¬ld¬z, A.O. Akdemir and E. Set, New Inequalities of Hadamard Type for Quasi-Convex Functions, AIP Conference Proceedings, 1470, 99-101 (2012); doi: 10.1063/1.4747649. [15] J.E. Peµcari´c, F. Porschan and Y.L. Tong, Convex Functions, Partial Orderings, and Statistical Applications, Academic Press Inc., 1992. [16] A. Saglam, M.Z. Sar¬kaya, H. Y¬ld¬r¬m, Some new inequalities of Hermite-Hadamard’s type, Kyung. Math. Jour., 50 (2010), 399-410. [17] M.Z. Sar¬kaya, N. Aktan, On the generalization some integral inequalities and their applications, Math. and Comp. Mod., 54(2011), 2175-2182. [18] S.H. Wang, B.-Y. Xi, F. Qi, Some new inequalities of Hermite-Hadamard type for n-time di¤erentiable functions which are m-convex, Analysis (Munich), 32 (2012), 247-262. [19] B.-Y. Xi, F. Qi, Some integral inequalities of Hermite-Hadamard type for convex functions with applications to means, J. Funct. Spaces Appl., 2012 (2012), http://dx.doi.org/10.1155/2012/980438. [20] Ç. Y¬ld¬z, Integral Inequalities for n Times Di¤erentiable Mappings, RGMIA Research Report Collection, 18 (2015), Article 5. [21] Ç. Y¬ld¬z, New Inequalities of the Hermite-Hadamard type for n-time di¤erentiable functions which are quasiconvex, Jour. Math. Ineq., (10) (3) (2016), 703-711. [22] Ç. Y¬ld¬z, M.E. Özdemir, New generalized inequalities of Hermite-Hadamard type for quasi-convex functions, AIP Conference Proceedings, 1726, 020053-1–020053-4 (2016); doi: 10.1063/1.4945879. ATATÜRK UNIVERSITY, K. K. EDUCATION FACULTY, DEPARTMENT OF MATHEMATICS, 25240, CAMPUS, ERZURUM, TURKEY E-mail address : [email protected] H GRADUATE SCHOOL OF NATURAL AND APPLI·ED SCIENCES, AGRI ¼ I·BRAHI·M ¼ ÇEÇEN UNIVERSITY, AGRI, TURKEY E-mail address : [email protected]