Inequalities for generalized Riemann–Liouville fractional integrals of generalized strongly convex functions

Some new integral inequalities for strongly (α,h –m)-convex functions via generalized Riemann–Liouville fractional integrals are established. The outcomes of this paper provide refinements of some fractional integral inequalities for strongly convex, stronglym-convex, strongly (α,m)-convex, and strongly (h –m)-convex functions. Also, the refinements of error estimations of these inequalities are obtained by using two fractional integral identities. Moreover, using a parameter substitution and a constant multiplier, k-fractional versions of established inequalities are also given.

Let f : I → R be a convex function and a, b ∈ I where a < b. Then the following inequality holds: If the function f is concave on I, then the inequality in (1.2) holds in reverse order. This inequality provides lower and upper estimates of the integral mean value of a convex function. The interest of many mathematicians was evoked by this inequality, and several generalizations, extensions, and variants of this inequality have been obtained. In the last two decades it has been remarkably studied, and a lot of papers have been published, see [1,11,13,14,17,24,[29][30][31][32][33][34] and the references therein. Fractional integral inequalities play an important role in mathematics as well as in other areas of mathematics because of their wide applications to establishing the uniqueness of solutions. These solutions can be obtained for certain fractional partial differential equations.
The main objective of this research is to obtain a few versions of the Hadamard inequality for generalized Riemann-Liouville fractional integrals. To achieve this goal, we employ the definition of strongly (α, hm) convex functions. The refinements of their error estimations are also established. Taking into account parameter substitution and a constant multiplier, k-fractional versions of Hadamard inequalities and their estimations for strongly (α, hm)-convex function are proved. In the course of this study, results obtained are a unification and generalization of the comparable results in the literature on Hadamard inequalities. Next, we give the definition of strongly (α, hm)-convex function as follows.
Definition 1 ([35]) Let J ⊆ R be an interval containing (0, 1), and let h : J → R be a nonnegative function. A function f : [0, b] → R is called strongly (α, hm)-convex function with modulus λ ≥ 0, if f is nonnegative and for all x, y ∈ [0, b], t ∈ (0, 1), m ∈ (0, 1], we have the inequality f xt + m(1t)y ≤ h t α f (x) + mh 1t α f (y)mλh t α h 1t α |y -x| 2 . (1.3) Fractional calculus is the study of derivatives and integrals of fractional order. Its history is nearly as old as the history of classical calculus. Nevertheless, it has gained the popularity and importance in extensive fields of science and engineering. This field has been widely adopted by many scholars. Recently, motivated by the classical Riemann-Liouville fractional integral operators, researchers have defined different integral operators, see [4,12,25]. The Riemann-Liouville fractional integral operator is defined as follows.
Definition 2 Let f ∈ L 1 [a, b]. Then left-sided and right-sided Riemann-Liouville fractional integrals of a function f of order μ, where (μ) > 0, are defined by (1.4) and Sarikaya et al. [27,28] elegantly obtained the following fractional integral inequalities of Hadamard type by using the Riemann-Liouville fractional integrals.
If f is a convex function on [a, b], then the following fractional integral inequality holds: with μ > 0.
, then the following fractional integral inequality holds: The definition of k-fractional integral is stated as follows.
Then k-fractional Riemann-Liouville integrals of order μ, where (μ) > 0, k > 0, are defined by (1.8) and where k (·) is defined as follows: The definition of generalized fractional integrals by a monotonically increasing function is given as follows.
The k-analogue of generalized fractional integrals is defined as follows. (1.13) Using the fact k (μ) = k μ k -1 ( μ k ) in (1.10) and (1.11) after replacing μ with μ k , we get For more details on the above defined fractional integrals, we refer the readers to see [21,26]. In the upcoming section, Hadamard inequalities for strongly (α, hm)-convex function via (1.10) and (1.11) are derived. Also, we give refinements of many fractional versions of Hadamard inequalities proved in [3, 5-10, 16, 18, 19, 27, 28]. In Sect. 3, by using two different fractional integral identities, error bounds of the established inequalities are given. Section 4 contains k-fractional versions of Hadamard inequalities and their estimations for strongly (α, hm)-convex function.

Fractional versions of Hadamard inequalities for strongly (α, h -m)-convex function
3). Then, by applying Definition 4 and multiplying by μ, we get the following inequality: Hence the first inequality of (2.1) is obtained. On the other hand, f is a strongly (α, hm)convex function with modulus c, we have the following inequality: Multiplying inequality (2.4) by t μ-1 and then integrating over the interval [0, 1], we get Using substitutions in (2.5) as considered in (2.3) leads to the second inequality of (2.1).
and if c = 0, then the result for (hm)-convex function can be obtained.
, then the following inequality for strongly mconvex function holds: and if c = 0 in the above inequality, then the result for m-convex function can be obtained.
Remark 2 For c > 0, all the results stated in the above corollaries and remark provide the refinements.

Theorem 5 Under the assumptions of Theorem 4, the following fractional integral inequal-
ity holds: 2), and integrating the resulting inequality over [0, 1] after multiplying by t μ-1 , we get (2.7), then by applying Definition 4 and multiplying by μ, we get the following inequality: Hence the first inequality of (2.6) is obtained. Since f is a strongly (α, hm)-convex function on [a, b] with modulus c, we have the following inequality: Multiplying (2.8) by t μ-1 and then integrating over [0, 1], we get Using the substitutions in (2.9) as considered in (2.7) leads to the second inequality of (2.6).
If c = 0 in the above inequality, then the result for (hm)-convex function can be obtained. Also, taking ψ as the identity function in the above inequality, the result for strongly (hm)convex function via Riemann-Liouville fractional integrals can be obtained.

Corollary 4
If α = 1 and h(t) = t in (2.6), then the following inequality holds for strongly m-convex function: If c = 0 in the above inequality, then the result for m-convex function can be obtained.
Remark 4 For c > 0, all the results stated in the above corollaries and remark provide the refinements.

Error estimations of Hadamard inequalities for strongly (α, h -m)-convex functions
This section is concerned with the error estimations of Hadamard inequalities for strongly (α, hm)-convex function using integral operators (1.10) and (1.11). The consequences of these inequalities reflect the refinements of error bounds of some fractional integral inequalities for convex, m-convex, (α, m)-convex, and (hm)-convex functions. We need the following identity to prove our next theorem.
Proof From Lemma 1, it follows that By using the strong (α, hm)-convexity of |f |, we have

For c = 0 in the above inequality, the result for (hm)-convex function can be obtained. Moreover, if ψ is an identity function, then the result for (hm)-convex function for
Riemann-Liouville fractional integrals can be obtained.

Corollary 6
If α = 1 and h(t) = t in (3.2), then the following inequality holds for strongly m-convex function: If c = 0 in the above inequality, then the result for m-convex function can be obtained.

Corollary 7
If α = μ = m = 1, h(t) = t and take ψ as the identity function in (3.2), we get the following inequality for strongly convex function: 2 32 .
To prove next two theorems, we need the following identity.  on (a, b). If [a, b] ⊂ Range(ψ) and m ∈ (0, 1], then the following fractional integral identity holds: Also, suppose that |f | q is a strongly (α, hm)-convex function on [a, b] for q ≥ 1, ψ is an increasing and positive monotone function on (a, b] having a continuous derivative ψ   on (a, b). If [a, b] ⊂ Range(ψ) and (α, m) ∈ (0, 1] 2 , then the following fractional integral inequality holds: Proof For q = 1, applying the Lemma 2 and using the strong (α, hm)-convexity of |f |, we have Now, for q > 1, we use Lemma 2 and the power mean inequality For c = 0 in the above inequality, the result for (hm)-convex function can be obtained. Moreover, if ψ is the identity function in the above inequality, then the result for (hm)convex function for Riemann-Liouville fractional integrals can be obtained.

Corollary 9
If α = 1 and h(t) = t in (3.6), then the following inequality holds for strongly m-convex function: If c = 0 in the above inequality, then the result for m-convex function can be obtained.

Corollary 10
If α = 1 in (3.7), then the following inequality holds for strongly (hm)convex function: For c = 0 in the above inequality, the result for (hm)-convex function can be obtained. Moreover, if ψ is the identity function in the above inequality, then the result for (α, hm)convex function for Riemann-Liouville fractional integrals can be obtained.

Corollary 11
If α = 1 and h(t) = t in (3.7), then the following inequality holds for strongly m-convex function: If c = 0 in the above inequality, then the result for m-convex function can be obtained.

k-Fractional versions of Hadamard inequalities for strongly (α, h -m)-convex function and their error estimations
In this section, we present k-fractional versions of Hadamard inequalities and their error estimations discussed in Sect. 2 and Sect. 3.