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Hermite-Hadamard Type Inequalities Obtained via Fractional Integral for Differentiable m-Convex and ([alpha], m)-Convex Functions.

1. Introduction and Preliminaries

The following definition is well known in the literature.

Definition 1. A function f : I [right arrow] R, [phi] = I [subset or equal to] R, is said to be convex on the interval I if the inequality

f(tx + (1 - t)y) [less than or equal to] tf(x) + (1 - t) f(y) (1)

holds for all x, y [member of] I and t [member of] [0, 1].

Geometrically, this means that if P, Q, and R are three points on the graph of f with Q between P and R, then Q is on or below the chord PR.

Theorem 2 (Hermite-Hadamard inequality). Let f : I [subset] R [right arrow] R be a convex function and a, b [member of] I with a < b. Then

f(a + b/2) [less than or equal to] 1/b - a [[integral].sup.b.sub.a] f(x) dx [less than or equal to] f(a) + f(b)/2. (2)

m-convexity was defined by Toader as follows.

Definition 3 (see [1]). The function f : [0, b] [right arrow] R, b > 0, is said to be m-convex, where m [member of] [0, 1], if one has

f(tx + m(1 - t)y) [less than or equal to] tf(x) + m(1 - t) f(y) (3)

for all x, y [member of] [0, b] and t [member of] [0, 1]. One says that f is m-concave if -f is m-convex. Denote by [K.sub.m](b) the class of all m-convex functions on [0, b] for which f(0) [less than or equal to] 0.

Obviously, for m = 1, Definition 3 recaptures the concept of standard convex functions on [a, b] and for m = 0 the concept of starshaped functions. The notion of m-convexity has been further generalized in [2] as it is stated in the following definition.

Definition 4 (see [2]). The function f : [0, b] [right arrow] R, b > 0, is said to be ([alpha], m)-convex, where ([alpha], m) [member of] [[0, 1].sup.2], if one has

f(tx + m(1 - t)y) [less than or equal to] [t.sup.[alpha]] f(x) + m(1 - [t.sup.[alpha]]) f (y) (4)

for all x, y [member of] [0, b] and t [member of] [0, 1].

Denote by [K.sup.[alpha].sub.m] (b) the class of all ([alpha], m)-convex functions on [0, b] for which f(0) [less than or equal to] 0.

It can be easily seen that when ([alpha], m) [member of] {(1, 1), (1, m)} one obtains the following classes of functions: convex and m-convex, respectively. Note that [K.sup.1.sub.1] (b) is a proper subclass of m-convex and ([alpha], m)-functions [0, b]. The interested reader can find more about partial ordering of convexity in [3].

Definition 5 (see [4]). Let f [member of] [L.sub.1] [a, b]. Then Riemann-Liouville integrals [mathematical expression not reproducible] of order [alpha] > 0 with a [greater than or equal to] 0 are defined by

[mathematical expression not reproducible], (5)

where

[GAMMA]([alpha]) = [[integral].sup.[infinity].sub.0] [e.sup.-x] [x.sup.[alpha]-1] dx (6)

is the Gamma function.

We now give the definition of the hypergeometric series which will be used in obtaining some integrals.

Definition 6 (see [4]). The integral representation of the hypergeometric functions is as follows:

[sub.2][F.sub.1] [a, b, c; z] = 1/B(b, c - b) [[integral].sup.1.sub.0] [t.sup.b-1] [(1 - t).sup.c-b-1] [(1 - zt).sup.-a] dt, (7)

where [absolute value of (z)] <1, c > b > 0, and

B(x, y) = [[integral].sup.1.sub.0] [t.sup.x-1] [(1 - t).sup.y-1] dt (8)

is Beta function with

B(x, y) = [GAMMA](x)[GAMMA](y)/[GAMMA](x + y). (9)

In the present paper, we establish some new Hermite-Hadamard's type inequalities for the classes of m-convex and ([alpha], m)-convex functions via Riemann-Liouville fractional integrals.

To prove our main results, we consider the following lemma.

Lemma 7 (see [5]). Let f: [a, b] [subset] R [right arrow] R be a differentiable function such that f' [member of] L[a, b], Then, for n [member of] N, k > 0, and x [member of] [a, b], one has

[mathematical expression not reproducible], (10)

where

[mathematical expression not reproducible]. (11)

2. Generalized Inequalities for m-Convex Functions

Theorem 8. Let I be on open real interval such that [0, [infinity]) [subset] I. Let f: I [right arrow] R be a differentiable function on I such that n [member of] N, k > 0 and f' [member of] L[a, b], where 0 [less than or equal to] a < b < [infinity]. If [absolute value of (f')] is an m-convex function on [a, b] for some fixed m [member of] (0, 1], then

[mathematical expression not reproducible], (12)

where x [member of] [a, b].

Proof. Using Lemma 7, taking modulus and the fact that [absolute value of (f')] is an m-convex function, we have

[mathematical expression not reproducible]. (13)

This completes the proof.

Remark 9. Observe that if in Theorem 8 we have m = n = 1, the statement of Theorem 8 becomes the statement of Theorem 1 in [6].

Theorem 10. Let I be on open real interval such that [0, [infinity]) [subset] I. Let f : I [right arrow] R be a differentiable function on I such that n [member of] N, k > 0 and f' [member of] L[a, b], where 0 [less than or equal to] a < b < [infinity]. If [[absolute value of (f')].sup.q] is an m-convex function on [a, b] for some fixed m [member of] (0, 1] and 1/p + 1/q = 1, q > 1, then

[mathematical expression not reproducible],

where x [member of] [a, b].

Proof. Using Lemma 7, Holder's inequality, and the fact that [[absolute value of (f')].sup.q] is an m-convex function,

[mathematical expression not reproducible]. (15)

This completes the proof.

Remark 11. Observe that if in Theorem 10 we have m = n = 1, the statement of Theorem 10 becomes the statement of Theorem 2 in [6].

Theorem 12. Let I be on open real interval such that [0, [infinity]) [subset] I. Let f : I [right arrow] R be a differentiable function on I such that n [member of] N, k > 0 and f' [member of] L[a, b], where 0 [less than or equal to] a < b < [infinity]. If [[absolute value of (f')].sup.q] is an m-convex function on [a, b] for some fixed m [member of] (0, 1] and q [greater than or equal to] 1, then

[mathematical expression not reproducible], (16)

where x [member of] [a, b].

Proof. Using Lemma 7, Power's mean inequality, and the fact that [[absolute value of (f')].sup.q] is an m-convex function,

[mathematical expression not reproducible]. (17)

This completes the proof.

Remark 13. Observe that if in Theorem 12 we have m = n = 1, the statement of Theorem 12 becomes the statement of Theorem 3 in [6].

3. Generalized Inequalities for ([alpha], m)-Convex Functions

Theorem 14. Let I be on open real interval such that [0, [infinity]) [subset] I. Let f : I [right arrow] R be a differentiable function on I such that n [member of] N, k > 0, and f' [member of] L[a, b], where 0 [less than or equal to] a < b < [infinity]. If [absolute value of (f')] is ([alpha], m)-convex function on [a, b] for some fixed ([alpha], m) [member of] [(0, 1].sup.2], then

[mathematical expression not reproducible], (18)

where x [member of] [a, b] and

[mathematical expression not reproducible]. (19)

Proof. Using Lemma 7 and taking modulus and the fact that [absolute value of (f')] is ([alpha], m)-convex function, we have

[mathematical expression not reproducible]. (20)

This completes the proof.

Remark 15. Observe that if in Theorem 14 we have [alpha] = 1, the statement of Theorem 14 becomes the statement of Theorem 8.

Theorem 16. Let I be on open real interval such that [0, [infinity]) [subset] I. Let f : I [right arrow] R be a differentiable function on I such that n [member of] N, k > 0, and f' [member of] L[a, b], where 0 [less than or equal to] a < b < [infinity]. If [[absolute value of (f')].sup.q] is ([alpha], m)-convex function on [a, b] for some fixed ([alpha], m) [member of] [(0, 1].sup.2] and 1/p + 1/q = 1, q > 1, then

[mathematical expression not reproducible], (21)

where x [member of] [a, b] and

c [(n + 1).sup.[alpha]] - [n.sup.[alpha]]/[alpha] + 1 D = [(n + 1).sup.[alpha]]. (22)

Proof. Using Lemma 7, Holder's inequality, and the fact that [[absolute value of (f')].sup.q] is ([alpha], m)-convex function,

[mathematical expression not reproducible]. (23)

This completes the proof.

Remark 17. Observe that if in Theorem 16 we have [alpha] = 1, the statement of Theorem 16 becomes the statement of Theorem 10.

Theorem 18. Let I be on open real interval such that [0, [infinity]) [subset] I. Let f : I [right arrow] R be a differentiable function on I such that n [member of] N, k > 0, and f' [member of] L[a, b], where 0 [less than or equal to] a < b < [infinity]. If [[absolute value of (f')].sup.q] is ([alpha], m)-convex function on [a, b] for some fixed ([alpha], m) [member of] [(0, 1].sup.2] and q [greater than or equal to] 1, then

[mathematical expression not reproducible], (24)

where A and B are given by (19) and x [member of] [a, b].

Proof. Using Lemma 7, Power's mean inequality, and the fact that [[absolute value of (f')].sup.q] is ([alpha], m)-convex function,

[mathematical expression not reproducible]. (25)

This completes the proof.

Remark 19. Observe that if in Theorem 18 we have [alpha] = 1, the statement of Theorem 18 becomes the statement of Theorem 12.

http://dx.doi.org/10.1155/2016/4765691

Competing Interests

The authors declare that they have no competing interests.

References

[1] G. Toader, "On a generalization of the convexity," Mathematica, vol. 30, no. 53, pp. 83-87, 1988.

[2] V. G. Mihesan, A Generalization of the Convexity, Seminar of Functional Equations, Approx. and Convex, Cluj-Napoca, Romania, 1993.

[3] J. E. Pecaric, F. Proschan, and Y. L. Tong, Convex Functions, Partial Orderings, and Statistical Applications, Academic Press, 1992.

[4] A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006.

[5] M. A. Noor, K. I. Noor, M. V. Mihai, and M. U. Awan, "Fractional Hermite-Hadamard inequalities for differentiable s-Godunova-Levin functions," Filomat, In press.

[6] M. V. Mihai and F.-C. Mitroi, "Hermite-Hadamard type inequalities obtained via Riemann-Liouville fractional calculus," Acta Mathematica Universitatis Comenianae, vol. 83, no. 2, pp. 209-215, 2014.

Erhan Set, (1) Suleyman Sami Karatas, (1) and Muhammad Adil Khan (2)

(1) Department of Mathematics, Faculty of Sciences, Ordu University, 52200 Ordu, Turkey

(2) Department of Mathematics, University of Peshawar, Peshawar, Pakistan

Correspondence should be addressed to Erhan Set; erhanset@yahoo.com

Received 1 July 2016; Accepted 20 September 2016

Academic Editor: Ahmed Zayed
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Title Annotation:Research Article
Author:Set, Erhan; Karatas, Suleyman Sami; Khan, Muhammad Adil
Publication:International Journal of Analysis
Article Type:Report
Date:Jan 1, 2016
Words:1850
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