New Hadamard-type inequalities for E-convex functions involving generalized fractional integrals

In this article, we establish some new Hadamard-type inequalities for E-convex functions involving generalized fractional integrals. These inequalities include a generalized Hadamard-type inequality and the corresponding right Hadamard-type inequalities for E-convex functions. The results presented here are generalizations of some of the results discussed in the recent literature.


Introduction and basic definitions
The theory of convexity is not only important in itself but also it contributes to almost all areas of mathematics. Convexity gives rise to inequalities, the Hadamard inequality is the first consequence of convex functions. The book by Hardy [1] has played a key role in popularizing the subject of convex analysis. Over the years, the idea of convex sets and convex functions has been largely generalized. Today, the study of convex functions has evolved into a broader theory of functions including quasiconvex functions [2,3], coordinated convex functions [4,5], preinvex functions [6], GA-convex functions [7], strongly convex functions [8], (g, ϕ h )-convex functions [9], E-convex functions [10] and so on. Youness [10] defined the E-convex set and the corresponding function as follows: Every convex function f on a convex set S is an E-convex function provided that E is an identity function. For a detailed explanation of E-convex functions see [10]. The Hadamard-type inequality for E-convex given in [11] is as follows: Theorem 1 Let E : J ⊂ R − → R be a continuous increasing function and ζ , η ∈ J with ζ < η. Let f : I ⊆ R − → R be an E-convex function on [ζ , η], then we have

.2) is Hadamard's inequality for E-convex functions.
Convexity is mixed with other mathematical concepts such as; optimization [12], time scale [13,14], quantum and postquantum calculus [15,16], and fractional calculus [3,11,[17][18][19]. Fractional calculus is basically a generalization of integer-order calculus. Strictly speaking, it is a generalization of operators beyond the integral order to real or complex order. Many fractional models have been proposed so far [20][21][22][23][24][25][26][27]. The key drivers behind such proposals are identified with the various real data corresponding to different systems under consideration requiring different kernels. Raina [27] and Agarwal [26] defined the following generalized fractional operators: Definition 3 Let f ∈ L(ζ , η), then for σ , ρ > 0, ω ∈ R the right-handed and left-handed generalized fractional integrals of f are, respectively, defined as follows: where F α σ ,ρ (s) is defined in [27] as follows: where R is a real positive constant. The coefficients α(n) (n ∈ N 0 = N ∪ {0}) are terms of a bounded sequence of positive real numbers and R is the set of real numbers. Moreover, the operators J α σ ,ρ,ζ +;ω f and J α σ ,ρ,η+;ω f are bounded on L(ζ , η), i.e., These fractional integrals are really important because of their generality. Many other fractional operators can be obtained by specifying the coefficients α(n). For instance, if we set n = 0, α(0) = 1 and ω = 0, we obtain the well-known Riemann-Liouville fractional operators and Lemma 1 ( [28,29]) For 0 < α ≤ 1 and 0 ≤ x < y, we have Fractional calculus has useful applications in almost all areas of applied mathematics and other sciences, see [30] and the references therein. In the present work, notions of E-convexity and generalized fractional operators are joined together. These ideas are independently utilized before, however, in combined form we obtain even more generalized results.

Main outcomes
In this section, mainly the Hadamard inequality for E-convex function (1.2) is extended using Definition 3 of generalized fractional integrals. Then, an identity is established for differentiable functions that is used to develop right Hadamard-type inequalities for the said extended Hadamard-type inequality. Likewise, another important identity is developed for twice-differentiable functions that is further used to develop more right Hadamardtype inequalities for the said extended Hadamard-type inequality for E-convex functions.
In the following, we use J to represent the interval of nonnegative real numbers and I to represent the interval of real numbers. Moreover, we use the following notations for brevity; for all σ , ρ ∈ R + and ω ∈ R.
On multiplying both sides of inequality and then integrating the resultant inequality with respect to t over [0, 1], we have Further suppose that u = tE(ζ ) + (1t)E(η) and v = (1t)E(ζ ) + tE(ζ ) and using the definition of generalized fractional integrals 2f Considering again the E-convexity of f over the interval [ζ , η], we have and on adding inequality (2.4) and inequality (2.5), we have On multiplying both sides of inequality (2.6) by , integrating with respect to t over the interval [0, 1] and finally using the definition of generalized fractional integrals, we have and then using the definition of generalized fractional integrals, we have On combining inequality (2.3) and inequality (2.7), we obtain the required result. Hence it is proved.
Remark 1 If in Theorem 2, the function E is chosen to be an identity function, then the following inequality holds for all σ , ρ ∈ R + and ω ∈ R: which was given in [31].
Remark 2 If in Theorem 2, the function E is chosen to be an identity function, α(0) = 1, ρ = λ and ω = 0, then the following inequality holds: which was given in [32].
the following identity holds for generalized fractional operators: Proof Solving the subsequent integral by integration by parts, then using a change of variable and finally the definition of the left generalized fractional integral operator Similarly, and on subtracting inequality (2.9) and inequality (2.10), then multiplying by and on submitting the expressions for I 1 and I 2 , we obtain the required result.
Hence it is proved.
Remark 3 If in Theorem 3, the function E is chosen to be an identity function, then the following inequality holds for all σ , ρ ∈ R + and ω ∈ R: which was given in [31].
Remark 4 If in Theorem 3, the function E is chosen to be an identity function, α(0) = 1, ρ = λ and ω = 0, then the following inequality holds: which was given in [32].
Theorem 4 Let E : J − → R be a continuous increasing function and ζ , η ∈ J with ζ < η.
Remark 5 If in Theorem 4, the function E is chosen to be an identity function, then the following inequality holds for all σ , ρ ∈ R + and ω ∈ R: , which was given in [33].
Hence it is proved.
Remark 7 If in Theorem 5, the function E is chosen to be an identity function, then the following inequality holds for all σ , ρ ∈ R + and ω ∈ R: , which was given in [33].
Remark 8 If in Theorem 5, the function E is chosen to be an identity function, α(0) = 1, ρ = λ and ω = 0, then the following inequality holds: which was given in [33].
Hence it is proved.
where α 5 is as defined in (2.19). Hence it is proved.
Remark 11 If in Theorem 7, the function E is chosen to be an identity function, then the following inequality holds for all σ , ρ ∈ R + and ω ∈ R: , which was given in [33].