Estimates of Upper Bound for Differentiable Functions Associated with k -Fractional Integrals and Higher Order Strongly s -Convex Functions

In this paper, we establish two integral identities associated with di ﬀ erentiable functions and the k -Riemann-Liouville fractional integrals. The results are then used to derive the estimates of upper bound for functions whose ﬁ rst or second derivatives absolute values are higher order strongly s -convex functions.


Introduction
Fractional calculus also known as noninteger calculus is a branch of mathematical analysis in which we discuss the integrals and derivatives of arbitrary order. The study of fractional calculus has a very long history, which can be traced back to the end of the 17th century; in 1695, L'Hospital wrote to Leibniz to discuss fractional derivative about a function. For hundreds of years, many mathematicians, such as Euler, Laplace, Fourier, Abel, Liouville, and Riemann, have carried out in-depth research on this subject (see [1]). Especially, in recent decades, the fractional calculus has found numerous applications in various fields such as mechanics, electricity, chemistry, biology, economics, notably control theory, and signal and image processing. Due to the backgrounds in practical applications, the fractional calculus has developed rapidly and has become a hot research topic (see [2][3][4]).
Among several known forms of fractional integrals, the Riemann-Liouville fractional integral has been investigated extensively, which is defined as follows: Definition 1 ( [3]). Let f ∈ L½a, b. The Riemann-Liouville integrals J α a+ f and J α b− f of order α > 0 with a ≥ 0 are defined by where is the gamma function.
In recent years, several researchers have utilized the concepts of fractional calculus to obtain the fractional analogues of classical inequalities. For example, Sarikaya et al. [5] established a generalized Hermite-Hadamard inequality via the Riemann-Liouville fractional integrals; Set [6] gave some fractional integral inequalities of the Ostrowski type through s-convex functions; Du et al. [7] deduced some variants of the fractional Hermite-Hadamard's inequality using the class of generalized (α,m)-preinvex functions; Noor et al. [8] used the class of the s-Godunova-Levin convex functions to obtain refinements of the fractional Hermite-Hadamard inequalities; Peng et al. [9] obtained the Riemann-Liouville fractional Simpson inequalities through generalized (m,h 1 ,h 2 )-preinvex functions; Wu and Awan [10] used the class of h-convex functions to derive the upper bound estimates of function involving fractional integrals; Wu et al. [11] established some fractional integral inequalities using k-th order differentiable strongly h-preinvex functions; Zhang et al. [12] provided some variations of the fractional Hermite-Hadamard's inequalities. For more results related to this topic, we refer the interested reader to [13][14][15][16][17] and references cited therein.
In [18], Mubeen and Habibullah introduced the k-fractional integral of the Riemann-Liouville type as follows: where is the k-gamma function.
Note that if k ⟶ 1, then the k-Riemann-Liouville fractional integrals reduces to the classical Riemann-Liouville fractional integral.
Sarikaya et al. [19,20] generalized the k-Riemann-Liouville fractional integrals and discussed their properties. Moreover, for the k-gamma function, they showed that Γ k ðαÞ = k α/k−1 Γðα/kÞ and Γ k ðα + kÞ = αΓ k ðαÞ. For k-beta function, it is defined by which implies that B k ðx, yÞ = ð1/kÞBððx/kÞ, ðy/kÞÞ and B k ðx, yÞ = Γ k ðxÞΓ k ðyÞ/Γ k ðx + yÞ: Motivated by the ideas of [19,20], in this paper, we first establish two identities for the k -Riemann-Liouville fractional integrals associated with differentiable functions. We then apply the results to derive some estimates of the upper bound for differentiable functions involving k-fractional integrals via higher order strongly s-convex functions.

Preliminaries and Lemmas
Let us briefly summarize the concepts on generalized convex functions which are related to the contents of this paper.
As a strengthening property of convexity, Polyak [21] introduced the strongly convex functions, as follows: Angulo et al. [22] presented an extension of strongly convex functions which is called strongly s-convex functions, i.e.: Here we provide a further extension of strongly s-convex functions, as follows: Remark 6. If σ = 2, then the class of strongly s-convex functions of order σ > 0 reduces to the class of strongly s-convex functions. For σ = 2 and s = 1, we have the class of classical strongly convex functions.
In the following, we establish two integral identities associated with differentiable functions and the k-Riemann-Liouville fractional integrals. These integral identities play important role in dealing with subsequent results.

Journal of Function Spaces
Proof. Let Integrating by parts gives Similarly, we have Using (11) and (12) in (10) leads to (9). This completes the proof of Lemma 7.

Estimates of the Upper Bound for Functions Whose First Derivatives Absolute Values Are Higher Order Strongly s-Convex Functions
Theorem 9. Let f : ½a, b ⊂ ð0, ∞Þ ⟶ ℝ be a differentiable function and f ′ ∈ L½a, b. If j f ′ j is strongly s-convex functions of order σ > 0, then for any α > 0, k > 0, μ > 0, s ∈ ð0, 1, and x ∈ ða, bÞ, we have where L 1 = akB k α + k, sk + k ð Þ+ x s + 1 , Proof. By Lemma 7 and the property of absolute value, we have Utilizing the fact that jf ′j is strongly s-convex functions of the order σ > 0, we obtain which implies the desired inequality (17). The proof of Theorem 9 is complete. is complete.
Proof. Using Lemma 7, Hölder's inequality, and the fact that j f ′j q is strongly s-convex functions of the order σ > 0, it follows that This completes the proof of Theorem 10.