Generalization of some fractional versions of Hadamard inequalities via exponentially (α; h − m)-convex functions

In this paper we give Hadamard inequalities for exponentially (α, h − m)-convex functions using Riemann-Liouville fractional integrals for strictly increasing function. Results for RiemannLiouville fractional integrals of convex, m-convex, s-convex, (α,m)-convex, (s,m)-convex, (h − m)convex, (α, h − m)-convex, exponentially convex, exponentially m-convex, exponentially s-convex, exponentially (s,m)-convex, exponentially (h − m)-convex, exponentially (α, h − m)-convex functions are particular cases of the results of this paper. The error estimations of these inequalities by using two fractional integral identities are also given.


Introduction
Fractional calculus is the study of integrals and derivatives of any arbitrary real or complex order. Its origin goes far back in 1695 when Leibniz and l'Hospital started discussion on the meaning of semi-derivative. The 17 th century witnessed the attention of this debate by many researchers working in the field of mathematics. Seeking in the formulation of fractional derivative/integral formulas Riemann and Liouville obtained their definitions with complementary functions. Later by Sonin and Letnikov along with others worked out the final form of fractional integral operator named Riemann-Liouville fractional integral/derivative operator, see [1,2]. After this definition of fractional integral/derivative operator the subject of fractional calculus become prominent in generalizing and extending the concepts of calculus and their applications. From the last few years, use of fractional calculus is also followed by scientists from various fields of engineering, sciences and economics. Some fractional integral operators have also been introduced in recent past see [3][4][5]. By applying fractional operators, extensive research has been carried out to establish various inequalities with motivating results, see [6][7][8][9]. Definition 1. [10] Let ϕ ∈ L 1 [ 1 , 2 ]. Then left-sided and right-sided Riemann-Liouville fractional integrals of a function ϕ of order ζ where (ζ) > 0 are given by and (1. 2) The k-analogue of the Riemann-Liouville fractional integral is defined as follows: Definition 2. [11] Let ϕ ∈ L 1 [ 1 , 2 ]. Then k-fractional Riemann-Liouville integrals of order ζ where (ζ) > 0, k > 0, are given by where Γ k (.) is defined by [12]: The generalized Riemann-Liouville fractional integrals via a monotonically increasing function are given as follows: [ 1 , 2 ]. Also let ψ be an increasing and positive monotone function on ( 1 , 2 ], further ψ has a continuous derivative ψ on ( 1 , 2 ). The left as well as right fractional integral operators of order ζ where (ζ) > 0 of ϕ with respect to ψ on [ 1 , 2 ] are given by

5)
and (1.6) The k-analogues of the above generalized Riemann-Liouville fractional integrals are defined as follows: Definition 4. [13] Let ϕ ∈ L 1 [ 1 , 2 ]. Also let ψ be an increasing and positive monotone function on ( 1 , 2 ], further ψ has a continuous derivative ψ on ( 1 , 2 ). Therefore left as well as right k-fractional integral operators of order ζ where (ζ) > 0 of ϕ with respect to ψ on [ 1 , 2 ] are given by and (1.8) For a detailed study of fractional integrals we refer the readers to [2,14]. Next, we give the definition of exponentially (α, h − m)-convex function as follows: Definition 5. [15] Let J ⊆ R be an interval containing (0, 1) and let h : J → R be a non-negative function. A function ϕ : (1.9) The above definition provides some kinds of exponential convexities as follows: Remark 1. (i) If we substitute α = 1 and h(t) = t s , then exponentially (s, m)-convex function in the second sense introduced by Qiang et al. in [16] can be obtained.
(ii) If we substitute α = m = 1 and h(t) = t s , then exponentially s-convex function introduced by Mehreen et al. in [17] can be obtained.
(iii) If we substitute α = m = 1 and h(t) = t, then exponentially convex function introduced by Awan et al. in [18] can be obtained.
In [19,20], the following Hadamard inequality for convex function ϕ : [ 1 , 2 ] → R is studied: (1.10) If this inequality holds in reverse order, then the function f is called concave function. This inequality was first published by Hermite in 1883 and later Hadamard proved it independently in 1893. Since its occurrence, it is in focus of researchers and had/has been studied for different kinds of convex functions. In past two decades it is generalized by using various types of fractional integral operators and authors have investigated a lot of versions of this inequality, see [21][22][23] with ζ > 0.
The following theorem gives an error estimation of the inequality (1.11).
, then the following fractional integral inequality holds: The k-analogues of Theorems 1 and 2 are given in the next two results.
, then the following inequality for k-fractional integral holds: with ζ > 0.
Theorem 5. [27] Under the assumptions of Theorem 4, the following inequality for k-fractional integral holds: An error estimation of the inequality (1.14) is given in the following theorem.

Main results
We give two fractional versions of the Hadamard inequality for exponentially (α, h − m)-convex functions, first one is given in the following theorem.
The following remark states the connection of Theorem 7 with already established results. In the following we give inequality (2.1) for exponentially (h − m)-convex, exponentially (s, m)convex, exponentially s-convex, exponentially m-convex and exponentially convex functions. Corollary 1. If we take α = 1 in (2.1), then the following inequality holds for exponentially (h − m)convex functions: Corollary 2. If we take α = 1 and h(t) = t s in (2.1), then the following inequality holds for exponentially (s, m)-convex functions: If we put m = 1 in the above inequality, then the result of exponentially s-convex function can be obtained.
Corollary 3. If we take α = 1 and h(t) = t in (2.1), then the following inequality holds for exponentially m-convex functions: Corollary 4. If we take α = m = 1 and h(t) = t in (2.1), then the following inequality holds for exponentially convex functions: The next theorem is another version of the Hadamard inequality for exponentially (α, h − m)-convex functions.
Theorem 8. Under the assumptions of Theorem 7, the following k-fractional integral inequality holds: (2.2) and integrating the resulting inequality over [0, 1] after multiplying with t ζ k −1 , we get Further, taking the maximum value of exponential function and using Definition 4, we get Again using exponentially (α, h − m)-convexity of ϕ, for t ∈ [0, 1], we have By integrating (2.10) over [0, 1] after multiplying with t ζ k −1 , the following inequality holds Again using substitutions as considered in (2.8), the second inequality of (2.7) can be obtained.
The following remark states the connection of Theorem 8 with already established results. In the following we give inequality (2.7) for exponentially (h − m)-convex, exponentially (s, m)convex, exponentially s-convex, exponentially m-convex and exponentially convex functions.
Corollary 5. If we take α = 1 in (2.7), then the following inequality holds for exponentially (h − m)convex functions: Corollary 6. If we take α = 1 and h(t) = t s in (2.7), then the following inequality holds for exponentially (s, m)-convex functions: If we put m = 1 in the above inequality, then the result of exponentially s-convex function can be obtained.

Error estimations of Hadamard inequalities for exponentially (α, h − m)-convex functions
In this section we give error estimations of the Hadamard inequalities by using exponentially (α, h − m)-convex functions via generalized Riemann-Liouville fractional integrals. The following identity is useful to prove the next theorem. Lemma 1.

Using (3.4) in (3.3), we get
The following remark states the connection of Theorem 9 with already established results. In the following we present the inequality (3.2) for exponentially (h − m)-convex, exponentially (s, m)-convex, exponentially s-convex, exponentially m-convex and exponentially convex functions.
Corollary 9. If we take α = 1 in (3.2), then the following inequality holds for exponentially (h − m)convex functions: Corollary 10. If we take α = 1 and h(t) = t s in (3.2), then the following inequality holds for exponentially (s, m)-convex functions: .
If we put m = 1 in the above inequality, then the result of exponentially s-convex function can be obtained.
Corollary 11. If we take α = m = 1 and h(t) = t in (3.2), then the following inequality holds for exponentially convex functions: Corollary 12. If we take α = m = k = 1 and h(t) = ψ(t) = t in (3.2), then the following inequality holds for exponentially convex functions via Riemann-Liouville fractional integrals: The following identity will be useful to obtain the next results.
The following remark states the connection of Theorem 10 with already established results. In the following we present the inequality (3.6) for exponentially (h − m)-convex, exponentially (s, m)-convex, exponentially s-convex, exponentially m-convex and exponentially convex functions.
Corollary 13. If we take α = 1 in (3.6), then the following inequality holds for exponentially (h − m)convex functions: Corollary 14. If we take α = 1 and h(t) = t s in (3.6), then the following inequality holds for exponentially (s, m)-convex functions: If we put m = 1 in the above inequality, then the result of exponentially s-convex function can be obtained.
Corollary 17. If we take α = 1 in (3.7), then the following inequality holds for exponentially (h − m)convex functions: If we put m = 1 in the above inequality, then the result of exponentially s-convex function can be obtained.
Corollary 19. If we take α = 1 and h(t) = t in (3.7), then the following inequality holds for exponentially m-convex functions: . Corollary 20. If we take α = m = 1 and h(t) = t in (3.7), then the following inequality holds for exponentially convex functions: In the following, we give inequality (3.7) for the operators given in (1.5) and (1.6).
Corollary 21. If we take k = 1 in (3.7), then the following inequality holds for exponentially convex functions:

Conclusions
The Hadamard inequalities presented in this work behave as generalized formulas which generate a number of fractional integral inequalities for all kinds of convex functions connected with exponentially (α, h − m)-convex function. Inequalities for Riemann-Liouville fractional integrals can also be deduced from the results of this paper. This work can be extended for other kinds of fractional integral operators exist in the literature.