On Strongly Convex Functions via Caputo–Fabrizio-Type Fractional Integral and Some Applications

Qi Li,Muhammad Shoaib Saleem,Peiyu Yan,Muhammad Sajid Zahoor,Muhammad Imran,Ahmet Ocak Akdemir

doi:10.1155/2021/6625597

Qi Li, Muhammad Shoaib Saleem + Show 4 more

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https://doi.org/10.1155/2021/6625597

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Abstract

The theory of convex functions plays an important role in the study of optimization problems. The fractional calculus has been found the best to model physical and engineering processes. The aim of this paper is to study some properties of strongly convex functions via the Caputo–Fabrizio fractional integral operator. In this paper, we present Hermite–Hadamard-type inequalities for strongly convex functions via the Caputo–Fabrizio fractional integral operator. Some new inequalities of strongly convex functions involving the Caputo–Fabrizio fractional integral operator are also presented. Moreover, we present some applications of the proposed inequalities to special means.

Highlights

E theory of convex functions plays an important role in the study of optimization problems. e fractional calculus has been found the best to model physical and engineering processes. e aim of this paper is to study some properties of strongly convex functions via the Caputo–Fabrizio fractional integral operator
It was observed by Caputo and Fabrizio in [8] that certain phenomena cannot be modelled by the already existing definition in the literature. at is why, they proposed a more general fractional derivative in [8] and named it as the Caputo–Fabrizio fractional integral operator
E theory of inequalities plays an important role in applied as well as in pure mathematics. e Hermite–Hadamard inequality is the most important inequality in the literature, and this inequality has been studied for different classes of convex functions, see [20,21,22,23,24]. e classical version of the Hermite–Hadamard inequality for convex functions is stated as follows: If ρ: I [a, b] ⊂ R ⟶ R is an integrable and continuous convex function, its mean value remains between the value of ρ at (a + b)/2 of interval I [a, b] and arithmetic mean value of ρ at the endpoints a, b ∈ I [a, b]

Summary

Introduction

Academic Editor: Ahmet Ocak Akdemir e theory of convex functions plays an important role in the study of optimization problems. e fractional calculus has been found the best to model physical and engineering processes. e aim of this paper is to study some properties of strongly convex functions via the Caputo–Fabrizio fractional integral operator. E aim of this paper is to study some properties of strongly convex functions via the Caputo–Fabrizio fractional integral operator. We present Hermite–Hadamard-type inequalities for strongly convex functions via the Caputo–Fabrizio fractional integral operator. Some new inequalities of strongly convex functions involving the Caputo–Fabrizio fractional integral operator are presented.

Results

Conclusion