Abstract

In this paper, we get the fractional integral inequalities obtained for geometric arithmetically (GA) convex functions by using functionals. The left hand side of the Hermite-Hadamard and Hermite-Hadamard-Fejér inequalities obtained by using Hadamard fractional integrals for Geometric Arithmetically-convex functions was obtained via functionals. We conclude that some results obtained in this paper are the refinements of the earlier results.

1. Introduction

Definition 1.1 A function is said to be convex if the inequality

is valid for all and . If this inequality reverses, then f is said to be concave on interval . This definition is well known in the literature.

It is well known that theory of convex sets and convex functions play an important role in mathematics and the other pure and applied sciences.

If is a convex function on the interval , then for any with we have the following double inequality

(1.1)

This double inequality is known in the literature as Hermite-Hadamard integral inequality for convex functions. Note that some of the classical inequalities for means can be derived from (1.1) for appropriate particular selections of the mapping . Both inequalities hold in the reversed direction if is concave. For some results which generalize, improve and extend the inequalities (1.1) we refer the reader to the recent papers (see ^{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}) and references therein.

Definition 1.2 ¹⁰ Let is said to be GA-convex (geometric-arithmetically convex) if

for all and . If this inequality is reversed, then is said to be geometric arithmetically concave.

In ², S. S. Dragomir proposed the following Hermite-Hadamard type inequalities which refine the first inequality of (1.1).

Theorem 1.1 ². Let is convex on . Then is convex, increasing on , and for all , we have

(1.2)

where

An analogous result for convex functions which refines the second inequality of (1.1) is obtained by G. S. Yang and M. C. Hong in ⁹ as follows.

Theorem 1.2 ⁹. Let is convex on . Then is convex, increasing on , and for all , we have

(1.3)

where

G. S. Yang and K. L. Tseng in ⁸ established some generalizations of (1.2) and (1.3) based on the following results.

Theorem 1.3 ⁸ Let be a convex function,

and let be defined by

for . Then is convex, increasing on and for all ,

The weighted generalization of Hermite-Hadamard inequality for GA-convex functions is as follows ¹⁶:

Theorem 1.4 Let be a GA-convex function, and . If is continuous and geometrically symmetric according to then

Specifically, if is taken in this theorem, the following Hermite-Hadamard inequality is obtained for GA-convex functions:

(1.4)

This inequality is also obtained in the case of specially in Theorem 3.1 and Theorem 3.3 in ¹⁴.

It is remarkable that M. Z. Sarıkaya et al. ⁷ proved the following interesting inequalities of Hermite-Hadamard type involving Riemann-Liouville fractional integrals.

Theorem 1.5 ⁷ Let be a positive function with and . If is a convex function on , then the following inequalities for fractional integrals hold:

(1.5)

with .

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

and

respectively. Here, is the Gamma function defined by

Definition 1.3 ¹² Let . The right-hand side and left-hand side Hadamard fractional integrals and of order with are defined by

and

respectively, where 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 (1.5).

Ruiyin Xiang ¹¹ proved the following Lemma and Theorem for interesting inequalities of Hermite-Hadamard type inequalities for convex functions via Riemann-Liouville fractional integrals.

Lemma 1.1 Let be a convex function and h be defined by

Then is convex, increasing on and for all ,

Theorem 1.6 Let be a positive function with and . If is a convex function on , then is convex and monotonically increasing on and

with , where

We can rewrite as follows the theorem in ¹³ for s = 1:

Theorem 1.7 Let . Then the following statements are true:

(1) The function is geometric-arithmetically convex on I if and only if is convex on the interval , where it is assumed that .

(2) If is decreasing and geometric-arithmetically convex on , then it is convex on

(3) If is increasing and convex on , then it is also geometric-arithmetically convex on .

For GA-convex functions Hermite-Hadamard inequalities obtained with the help of fractional integrals can be given as follows ¹⁵.

Theorem 1.8 Let be a function, , and . If the function is GA-convex on , then the following inequality for the fractional integrals hold:

(1.6)

with .

We have obtained the left sides of the (1.4) and (1.6) inequalities using a functional that we have defined. We have also obtained the left side of the Hermite-Hadamard-Fejér inequality through Hadamard fractional integrals for GA-convex functions.

2. The Left Hand Sides of the Hermite-Hadamard and Hermite-Hadamard-Fejér Inequalities via Functionals

Theorem 2.1 Let a be geometric arithmetically (GA)-convex function, ,

and let be defined by

for . Then h is convex, increasing on and for all ,

Proof. We note that if is GA-convex according to Theorem 1.7 and is linear, then the composition is convex on . According to the Theorem 1.3, also we note that a positive constant multiple of a convex function and a sum of two convex functions are convex, hence function

is convex and increasing on and for all ,

that is,

Lemma 2.1 Let be a GA-convex function and be defined by

Then is convex, increasing on and for all

Proof. If is geometric-arithmetically (GA) convex, then according to the Theorem 1.7 the function is convex on . Then the function defined by

is convex and increasing on and for

If substituting in the above inequality, we have

Theorem 2.2 Let be a function and . If the function is GA-convex on , then defined by

is convex and monotonically increasing on and

Proof. i) Firstly, let . We need to show that

Using the definition of , we can write the following

Since the function is geometric-arithmetically convex, we get

So

from which we get is convex on .

ii) By elementary calculus, we have

It follows from Lemma 2.1 that function is monotonically increasing on Since is nonnegative, hence is increasing on . Finally, from

and

we have completed the proof.

Corollary 2.1 With assumptions in Theorem 2.2, if , we get

where the function is defined as Theorem 2.2, which is just the result in Theorem 2.2.

Remark 2.1 The inequality obtained in Theorem 3.2 gives us the left side of the inequality obtained in Theorem 1.8.

The next theorem is a generalization of Theorem 2.2:

Theorem 2.3 Let be a function and . If the function is GA-convex on and is integrable, nonnegative and symmetrically according to that is for all , then defined by

is convex and monotonically increasing on , for

Proof. i) Firstly, let , , . We need to show that

Using the definition of we can write the following

So,

from which we get is convex on .

ii) By elementary calculus, we have

Since the function is simmetrically according to the

So,

It follows from Lemma 2.1 that

is monotonically increasing on . Since

is nonnegative, hence is increasing on . Finally, from

Here, by an easy calculation we get

and

This completes the proof of Theorem.

Remark 2.2. If in Theorem 2.3, then the following equality holds:

3. Conclusion

In this paper, we obtain some new Hermite-Hadamard and Hermite-Hadamard-Fejér type inequalities for geometric arithmetically convex functions via Hadamard 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 left side hand of Hermite-Hadamard and Hermite-Hadamard-Fejér inequalities for geometric arithmetically convex functions via Hadamard fractional integrals.

References