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

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.

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On Caputo–Fabrizio Fractional Integral Inequalities of Hermite–Hadamard Type for Modified h -Convex Functions

Journal of Mathematics ◽

10.1155/2020/8829140 ◽

2020 ◽

Vol 2020 ◽

pp. 1-17

Author(s):

Xiaobin Wang ◽

Muhammad Shoaib Saleem ◽

Kiran Naseem Aslam ◽

Xingxing Wu ◽

Tong Zhou

Keyword(s):

Integral Operator ◽

Fractional Calculus ◽

Convex Function ◽

Convex Functions ◽

Fractional Derivatives ◽

Fractional Integral ◽

Applied Mathematics ◽

Fractional Integral Operator ◽

Real Numbers ◽

Special Means

The theory of convex functions plays an important role in engineering and applied mathematics. The Caputo–Fabrizio fractional derivatives are one of the important notions of fractional calculus. The aim of this paper is to present some properties of Caputo–Fabrizio fractional integral operator in the setting of h -convex function. We present some new Caputo–Fabrizio fractional estimates from Hermite–Hadamard-type inequalities. The results of this paper can be considered as the generalization and extension of many existing results of inequalities and convex functions. Moreover, we also present some application of our results to special means of real numbers.

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On Generalized Strongly Convex Functions and Unified Integral Operators

Mathematical Problems in Engineering ◽

10.1155/2021/6695781 ◽

2021 ◽

Vol 2021 ◽

pp. 1-16

Author(s):

Timing Yu ◽

Ghulam Farid ◽

Kahkashan Mahreen ◽

Chahn Yong Jung ◽

Soo Hak Shim

Keyword(s):

Integral Operator ◽

Convex Function ◽

Convex Functions ◽

Fractional Integral ◽

Integral Operators ◽

Integral Inequalities ◽

Operator Inequalities ◽

Strongly Convex Functions ◽

Strongly Convex ◽

Left And Right

In this paper, we define a strongly exponentially α , h − m -convex function that generates several kinds of strongly convex and convex functions. The left and right unified integral operators of these functions satisfy some integral inequalities which are directly related to many unified and fractional integral inequalities. From the results of this paper, one can obtain various fractional integral operator inequalities that already exist in the literature.

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Hermite–Hadamard-Type Inequalities for Generalized Convex Functions via the Caputo-Fabrizio Fractional Integral Operator

Journal of Function Spaces ◽

10.1155/2021/5640822 ◽

2021 ◽

Vol 2021 ◽

pp. 1-8

Author(s):

Dong Zhang ◽

Muhammad Shoaib Saleem ◽

Thongchai Botmart ◽

M. S. Zahoor ◽

R. Bano

Keyword(s):

Integral Operator ◽

Convex Functions ◽

Fractional Integral ◽

Fractional Integral Operator ◽

Generalized Convex Functions ◽

Nonconvex Functions ◽

New Inequalities

Due to applications in almost every area of mathematics, the theory of convex and nonconvex functions becomes a hot area of research for many mathematicians. In the present research, we generalize the Hermite–Hadamard-type inequalities for p , h -convex functions. Moreover, we establish some new inequalities via the Caputo-Fabrizio fractional integral operator for p , h -convex functions. Finally, the applications of our main findings are also given.

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A note on fractional integral operator associated with multiindex Mittag-Leffler functions

Filomat ◽

10.2298/fil1607931c ◽

2016 ◽

Vol 30 (7) ◽

pp. 1931-1939 ◽

Cited By ~ 19

Author(s):

Junesang Choi ◽

Praveen Agarwal

Keyword(s):

Integral Operator ◽

Fractional Calculus ◽

Fractional Integral ◽

Fractional Integration ◽

Fractional Integral Operator ◽

Integration Formula ◽

Special Cases

Recently Kiryakova and several other ones have investigated so-called multiindex Mittag-Leffler functions associated with fractional calculus. Here, in this paper, we aim at establishing a new fractional integration formula (of pathway type) involving the generalized multiindex Mittag-Leffler function E?,k[(?j,?j)m;z]. Some interesting special cases of our main result are also considered and shown to be connected with certain known ones.

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New Modifications of Integral Inequalities via ℘-Convexity Pertaining to Fractional Calculus and Their Applications

Mathematics ◽

10.3390/math9151753 ◽

2021 ◽

Vol 9 (15) ◽

pp. 1753

Author(s):

Saima Rashid ◽

Aasma Khalid ◽

Omar Bazighifan ◽

Georgia Irina Oros

Keyword(s):

Integral Operator ◽

Convex Functions ◽

Hypergeometric Functions ◽

Fractional Integral ◽

Type Inequality ◽

Fractional Integral Operator ◽

Integral Inequalities ◽

Convex Mappings ◽

Special Cases ◽

Harmonically Convex Functions

Integral inequalities for ℘-convex functions are established by using a generalised fractional integral operator based on Raina’s function. Hermite–Hadamard type inequality is presented for ℘-convex functions via generalised fractional integral operator. A novel parameterized auxiliary identity involving generalised fractional integral is proposed for differentiable mappings. By using auxiliary identity, we derive several Ostrowski type inequalities for functions whose absolute values are ℘-convex mappings. It is presented that the obtained outcomes exhibit classical convex and harmonically convex functions which have been verified using Riemann–Liouville fractional integral. Several generalisations and special cases are carried out to verify the robustness and efficiency of the suggested scheme in matrices and Fox–Wright generalised hypergeometric functions.

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Some Inequalities of Generalized p-Convex Functions concerning Raina’s Fractional Integral Operators

Journal of Mathematics ◽

10.1155/2021/3089553 ◽

2021 ◽

Vol 2021 ◽

pp. 1-9

Author(s):

Changyue Chen ◽

Muhammad Shoaib Sallem ◽

Muhammad Sajid Zahoor

Keyword(s):

Integral Operator ◽

Convex Function ◽

Convex Functions ◽

Fractional Integral ◽

Applied Mathematics ◽

Optimization Theory ◽

Integral Operators ◽

Fractional Integral Operator ◽

Fractional Integral Operators

Convex functions play an important role in pure and applied mathematics specially in optimization theory. In this paper, we will deal with well-known class of convex functions named as generalized p-convex functions. We develop Hermite–Hadamard-type inequalities for this class of convex function via Raina’s fractional integral operator.

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New Hadamard-type inequalities via $(s,m_{1},m_{2})$-convex functions

Vestnik Udmurtskogo Universiteta Matematika Mekhanika Komp yuternye Nauki ◽

10.35634/vm210405 ◽

2021 ◽

Vol 31 (4) ◽

pp. 597-612

Author(s):

B. Bayraktar ◽

S.I. Butt ◽

Sh. Shaokat ◽

J.E. Nápoles Valdés

Keyword(s):

Integral Operator ◽

Convex Functions ◽

Fractional Integral ◽

Fractional Integral Operator ◽

Integral Inequalities ◽

Class Of Functions

The article introduces a new concept of convexity of a function: $(s,m_{1},m_{2})$-convex functions. This class of functions combines a number of convexity types found in the literature. Some properties of $(s,m_{1},m_{2})$-convexities are established and simple examples of functions belonging to this class are given. On the basis of the proved identity, new integral inequalities of the Hadamard type are obtained in terms of the fractional integral operator. It is shown that these results give us, in particular, generalizations of a number of results available in the literature.

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Some Estimates for Generalized Riemann-Liouville Fractional Integrals of Exponentially Convex Functions and Their Applications

Mathematics ◽

10.3390/math7090807 ◽

2019 ◽

Vol 7 (9) ◽

pp. 807 ◽

Cited By ~ 22

Author(s):

Saima Rashid ◽

Thabet Abdeljawad ◽

Fahd Jarad ◽

Muhammad Aslam Noor

Keyword(s):

Convex Function ◽

Convex Functions ◽

Fractional Integral ◽

Fractional Integrals ◽

Order Derivative ◽

Fundamental Identity ◽

First Order ◽

Special Means ◽

Exponentially Convex Functions ◽

New Inequalities

In the present paper, we investigate some Hermite-Hadamard ( HH ) inequalities related to generalized Riemann-Liouville fractional integral ( GRLFI ) via exponentially convex functions. We also show the fundamental identity for GRLFI having the first order derivative of a given exponentially convex function. Monotonicity and exponentially convexity of functions are used with some traditional and forthright inequalities. In the application part, we give examples and new inequalities for the special means.

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