REFINEMENTS OF HERMITE-HADAMARD TYPE INEQUALITIES FOR CONVEX FUNCTIONS VIA FRACTIONAL INTEGRALS

Abstract

In this note, two new mappings associated with convexity are propoesd, by which we obtain some new Hermite-Hadamard type inequalities for convex functions via Riemann-Liouville fractional integrals. We conclude that the results obtained in this work are the refinements of the earlier results.

1. Introduction

If f : I → R is a convex function on the interval I, then for any a, b ∈ I with a ≠ b we have the following double inequality

This remarkable result is well known in the literature as the Hermite-Hadamard inequality.

Since then, some refinements of the Hermite-Hadamard inequality for convex functions have been extensively obtained by a number of authors (e.g., [1], [2], [3], [4], [5], [6], [7], [8], [9] and [10]).

In [4], S. S. Dragomir proposed the following Hermite-Hadamard type in-equalities which refine the first inequality of (1).

Theorem 1.1 ([4]). Let f is convex on [a, b]. Then H is convex, increasing on [0, 1], and for all t ∈ [0, 1], we have

where

An analogous result for convex functions which refines the second inequality of (1) is obtained by G. S. Yang and M. C. Hong in [13] as follows.

Theorem 1.2 ([13]). Let f is convex on [a, b]. Then P is convex, increasing on [0, 1], and for all t ∈ [0, 1], we have

where

G. S. Yang and K. L. Tseng in [12] established some generalizations of (2) and (3) based on the following results.

Theorem 1.3 ([12]). Let f : [a, b] → R be a convex function, 0 < α < 1, and let h be defined by h(t) = (1 − β)f(A − βt) + βf(A + (1 − β)t), t ∈ [0, u0]. Then h is convex, increasing on [0, u0] and for all t ∈ [0, u0],

It is remarkable that M. Z. Sarikaya et al. [11] proved the following interesting inequalities of Hermite-Hadamard type involving Riemann-Liouville fractional integrals.

Theorem 1.4 ([11]). Let f : [a, b] → R be a positive function with a < b and f ∈ L1[a, b]. If f is a convex function on [a, b], then the following inequalities for fractional integrals hold:

with α > 0.

We remark that the symbols denote the left-sided and right-sided Riemann-Liouville fractional integrals of the order α ≥ 0 with a ≥ 0 which are defined by

and

respectively. Here, Г(α) is the Gamma function defined by

In this paper, we establish some new Hermite-Hadamard type inequalities for convex functions via Riemann-Liouville fractional integrals which refine the inequalities of (4).

 

2. Main results

Lemma 2.1. Let f : [a, b] → R be a convex function and h be defined by

Then h(t) is convex, increasing on [0, b − a] and for all t ∈ [0, b − a],

Proof. We can obtain the result by taking in Theorem 1.3. □

Theorem 2.2. Let f : [a, b] → R be a positive function with a < b and f ∈ L1[a, b]. If f is a convex function on [a, b], then WH is convex and monotonically increasing on [0, 1] and

with α > 0, where

Proof. Firstly, let t1, t2, β ∈ [0, 1], then

Since f is convex, we get

So

from which we get W H is convex on [0, 1]. Next, by elementary calculus, we have

It follows from Lemma 2.1 that is increasing on [0, b−a]. Since is nonnegative, hence W H(t) is increasing on [0, 1]. Finally, from

and

we have completed the proof. □

Similarly, we have the following theorem:

Theorem 2.3. Let f be defined as in Theorem 2.2, then W P is convex and monotonically increasing on [0, 1] and

with α > 0, where

Proof. We note that if f is convex and g is linear, then the composition f ◦ g is convex. Also we note that a positive constant multiple of a convex function and a sum of two convex functions are convex, hence are convex, from which we get that W P(t) is convex. Next, by elementary calculus, we have

It follows from Lemma 2.1 that and k(t) = b − a − (1 − t)x are increasing on [0, b − a] and [0, 1], respectively. Hence is increasing on [0, 1]. Since is nonnegative, it follows that W P is monotonically increasing on [0, 1]. Finally, from

and

we get the desired result. □

Corollary 2.4. With assumptions in Theorem 2.2, if α = 1, we get

where H(t) is defined as Theorem 1.1, which is just the result in Theorem 1.1.

Corollary 2.5. With assumptions in Theorem 2.3, if α = 1, we get

where P(t) is defined as Theorem 1.2, which is just the result in Theorem 1.2.

 

3. Conclusion

In this note, we obtain some new Hermite-Hadamard type inequalities for convex functions via Riemann-Liouville fractional integrals. We conclude that the results obtained in this work are the refinements of the earlier results. An interesting topic is whether we can use the methods in this paper to establish the Hermite-Hadamard inequalities for convex functions on the co-ordinates via Riemann-Liouville fractional integrals.