On exponentially (h1, h2)-convex functions and fractional integral inequalities related

In this work the concept of exponentially (h1, h2)-convex function is introduced and using it, the Hermite-Hadamard inequality and some bounds for the right side of this inequality, via Raina’s fractional integral operator and generalized convex functions, are established.


Introduction
In many practical investigations it is necessary to bound one quantity by another. The classical inequalities are very useful for this purpose. An enormous amount of efforts has been devoted to the extension of the classical inequalities and to the applications of the same in diverse areas of science: estimation of integrals, special functions of mathematical physics, electrostatic field and capacitance, signal analysis, dynamical system stability and control and others.
One of the most discussed inequalities in recent work is the classic Hermite-Hadamard inequality. In [7], J. Hadamard stated his famous inequality in this way.
Theorem 1. Let f be a convex function over [a, b] , a < b. If f is integrable over [a, b] , then . tives of any arbitrary real or complex order) has gained considerable popularity and importance during almost the past five decades or so, due mainly to its demonstrated applications in numerous seemingly diverse and widespread fields of science and engineering. It does indeed provide several potentially useful tools for solving differential and integral equations, and various other problems involving special functions of mathematical physics as well as their extensions and generalizations in one and more variables.
The inequalities involving more general fractional integral operators have also been considered in [2,12,16,19]. Since work in this direction has received a lot of attention, as evidenced in the work of S. Turhan et. al. [13,20] and J. E. Hernández Hernández and M. J. Vivas-Cortez [8,9,10,21], in this work we establish a general expression of some Hermite-Hadamard type inequa-lities by the introduction of the concept of exponentially (h 1 , h 2 )−convex function and using the Raina's fractional integral operator.
It is easy to verify that J σ ρ,λ,a+;w ϕ and J σ ρ,λ,b−;w ϕ are bounded integral Many useful fractional integral operators can be obtained by specializing the coefficient σ(k). By example, the classical Riemann-Liouville fractional integrals I α a+ and I α b− of order α and follow from (3) and (4) setting λ = α, σ(0) = 1 and w = 0. The Hermite-Hadamard integral inequality for the Raina's fractional integral operator is established in [22] as follows.
Theorem 2. Let λ ∈ R + , a, b ∈ R, a < b and φ : [a, b] → R be a convex function. Then 2.2. About Generalized convexity. The well known concept of convex function is due to W. Jensen and it is established as follow.
Definition 1. A function f : I ⊂ R → R is called convex on the interval I, if the following inequality holds for all a, b ∈ I and t ∈ [0, 1].
From the work of S. S. Dragomir et. al. [5], we extract the following definition.
Definition 2. Let f : I ⊂ R → R be a non-negative function where I is an interval. It is said that f belongs to the class P (I) or f is a P −convex if for all a, b ∈ I and t ∈ [0, 1] the following inequality holds Also, H. Hudzik and L. Maligranda, in [11], disused about some properties of the following generalized concept of convexity.
for all a, b ∈ I and t ∈ [0, 1]. This is denoted by f ∈ K 1 s . A function f : R + → R, is said to be s-convex in the second sense if for all a, b ∈ I and t ∈ [0, 1]. This is denoted by f ∈ K 2 s . The first class of functions in Definition 3 were introduced by Orlicz W. in [15], and the second class by W. W. Breckner in [3].
G. Cristescu et. al., in [4], in order to study bounds of the second degree cumulative frontier gaps of functions with generalized convexity functions, introduced the so-called (h 1 , h 2 )−convex functions.
holds for all a, b ∈ I and t ∈ [0, 1]. The functions that transform the inequality in an equality is called (h 1 , h 2 )−affine function.
, then we have the s−convexity in the first sense, and If h 1 (t) = t s and h 2 (t) = (1 − t) s for all t ∈ [0, 1], we get the s−convexity in the second sense.
The exponentially convex functions are of interest for the development of this work. In the works of T. Antczac [1] and S. S. Dragomir [6] introduce this concept and find some results related to the Hermite-Hadamard inequality.

Definition 5.
A positive function f : I → R is said to be an exponentially convex function if the inequality holds for all a, b ∈ I and t ∈ [0, 1].

Main Results
A positive function f : I → R, where I is an interval include in R, is called exponentially (h 1 , h 2 ) −convex if the following inequality holds for all x, y ∈ I and t ∈ [0, 1] Remark 2. Note that: (1) If h 1 = h 2 ≡ 1 then we have an exponentially P −convex function.
then we have the exponentially s−convexity in the first sense.
we get the exponentially s−convexity in the second sense.
First, we establish the Hermite-Hadamard inequality for exponentially convex function using Raina's fractional integral operator.
k=0 a sequence of nonnegatives real numbers. Let f : [a, b] → R be an exponentially (h 1 , h 2 )−convex function, then the following inequalities holds Proof. Note that .
in both sides of the above inequality Integrating over t ∈ [0, 1] we have With a convenient change of variable we have By replacement of (6) and (7) in (5) we have For the right side of the proposed inequality we have Adding these inequalities and integrating over t ∈ [0, 1] we obtain With the change of variable u = ta + (1 − t)b and v = tb + (1 − t)a in the first integral of the above inequality it is obtained and letting it is attained the desired result.
The following Lemma will be useful to establish some others inequalities related with the right side of the Hermite-Hadamard inequality for exponentially convex functions using the Raina's fractional integral operator. Lemma 3.1. Let λ, ρ > 0, w ∈ R, and σ = {σ(k)} ∞ k=0 a sequence of nonnegatives real numbers. Let f : [a, b] → R be a differentiable mapping on (a, b) with a < b and λ > 0. If e f ∈ L 1 ([a, b]) then the following equality for the Raina's fractional integral operator holds Proof. Using integration by parts it follows that and Subtracting I 2 from I 1 it is attained the desired result.  L 1 ([a, b]) and |f | is (g 1 , g 2 ) −convex then the following inequality for the Raina's fractional integral operator holds Proof. Using the Lemma 3.1 and the triangular inequality we have Now, we discuse the integrals involve in (8) using the exponentially (h 1 , h 2 )− convexity of f and the (g 1 , g 2 )−convexity of |f |. First, Similarly, for the second integral we have By replacement of (9) and (10) in (8) then it follows the result.
Using the previous Theorem some Corollary is deduced. Corollary 1. Let λ, ρ > 0, w ∈ R, and σ = {σ(k)} ∞ k=0 a sequence of nonnegatives real numbers. Let f : [a, b] → R be a differentiable mapping on (a, b) with a < b, and exponentially convex. If e f ∈ L 1 ([a, b]) and |f | is a convex function then the following inequality for the Raina's fractional integral operator holds
Making the corresponding substitutions in Theorem 4 it follows the desired result.
Corollary 2. Let λ, ρ > 0, w ∈ R, and σ = {σ(k)} ∞ k=0 a sequence of nonnegatives real numbers. Let f : [a, b] → R be a differentiable mapping on (a, b) with a < b, and exponentially convex. If e f ∈ L 1 ([a, b]) and |f | is convex and bounded by some M > 0, then the following inequality for the Raina's fractional integral operator holds Proof. Noting that then, using Corollary 1 easily it finds the result.

Note that
The proof is complete.
Corollary 4. Let λ, ρ > 0, w ∈ R, and σ = {σ(k)} ∞ k=0 a sequence of nonnegatives real numbers. Let f : [a, b] → R be a differentiable mapping on (a, b) with a < b, and exponentially convex. If e f ∈ L 1 ([a, b]) and |f | is a s−convex function in the second sense then the following inequality for the Raina's fractional integral operator holds for k = 0, 1, 2, . . .
By replacement of these values in Theorem 4 it is attained the result.
Remark 4. Since in the preliminary section is mentioned the fact that from the Raina's fractional integral the fractional integrals of Riemann-Liouville and the classic integral of Riemman can be deduced then the results found in Theorem 4 and Corollaries 1,2, 3 and 4 are useful to express them in terms of these integrals.

Conclusion
In the present work we established the Hermite-Hadamard inequality for exponentially convex functions using the Raina's fractional integral and from this result we deduced some results found in [17,18]. Also from Lemma 3.1 it was established a general theorem from which some fractional integral inequalities for exponentially convex functions, exponentially P −convex functions and exponentially s−convex functions in the second sense were found.
The usefulness of the theorems presented and the proposed technique can be applied to other types of generalized convex functions, for example, M T −convex functions [14].