Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 18 (2023), 223 -- 257

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This work is licensed under a Creative Commons Attribution 4.0 International License.

A COMPREHENSIVE REVIEW OF THE HERMITE-HADAMARD INEQUALITY PERTAINING TO FRACTIONAL DIFFERENTIAL OPERATORS

Muhammad Tariq, Sotiris K. Ntouyas, Asif Ali Shaikh and Jessada Tariboon

Abstract. A review on Hermite-Hadamard type inequalities connected with a different classes of convexities and fractional differential operators is presented. In the various classes of convexities it includes, classical convex functions, quasi-convex functions, p-convex functions, strongly-m-convex functions, strongly-(θ,m)-convex functions, (s, m)-convex functions, (θ, h- m)-convex functions, strongly (θ, h- m)-convex functions, (h, m)-convex functions of the second type, m-convex functions, h-convex functions, (h,m)-convex functions, relative-convex functions, exponentially (θ, h- m)-convex functions, harmonically h-convex functions and geometric-arithmetically s-convex functions. In the fractional differential operators it includes, Caputo fractional derivative, k-Caputo fractional derivative and Hilfer fractional derivative.

2020 Mathematics Subject Classification: 26A51; 26A33; 26D07; 26D10; 26D15
Keywords: Hermite-Hadamard inequalities; convex function; Caputo fractional derivative; k-Caputo fractional derivative; Hilfer fractional derivative

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Muhammad Tariq
Department of Basic Sciences and Related Studies,
Mehran University of Engineering and Technology,
Jamshoro 76062, Pakistan.
e-mail: captaintariq2187@gmail.com

Sotiris K. Ntouyas
Department of Mathematics, University of Ioannina,
451 10 Ioannina, Greece
e-mail: sntouyas@uoi.gr


Asif Ali Shaikh
Department of Basic Sciences and Related Studies,
Mehran University of Engineering and Technology,
Jamshoro 76062, Pakistan.
and
Department of Mathematics, Near East University,
99138 Mersin, Turkey.
e-mail: asif.shaikh@faculty.muet.edu.pk

Jessada Tariboon
Intelligent and Nonlinear Dynamic Innovations, Department of Mathematics,
Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok,
Bangkok 10800, Thailand.
e-mail: jessada.t@sci.kmutnb.ac.th




http://www.utgjiu.ro/math/sma