Naslov
Analytical Inequalities for Fractional Calculus Operators and the Mittag-Leffler Function

Autori
Andrić, Maja ; Farid, Ghulam ; Pečarić, Josip

Vrsta, podvrsta i kategorija knjige
Autorske knjige, monografija, znanstvena

Izdavač
Element

Grad
Zagreb

Godina
2021

Stranica
272

ISBN
978-953-197-813-2

Ključne riječi
fractional calculus ; integral inequality ; Mittag-Leffler function

Sažetak
Fractional calculus is a theory of differential and integral operators of non-integer order. In recent years, considerable interest in this theory has been stimulated due to its many applications in almost all applied science, especially in numerical analysis and various fields of physics and engineering. Inequalities which involve integrals of functions and their derivatives, whose study has a history of about a century, are of great importance in mathematics, with far-reaching applications in the theory of differential equations, approximations and probability, among others. They occupy a central place in mathematical analysis and its applications. Fractional differentiation inequalities have applications to fractional differential equations ; the most important ones are in establishing uniqueness of the solution of initial problems and giving upper bounds to their solutions. These applications have motivated many researchers in the field of integral inequalities to investigate certain extensions and generalizations using different fractional differential and integral operators. There are several well-known forms of fractional operators: Riemann-Liouville, Weyl, Erdelyi-Kober, Hadamard, Katugampola are just a few. As a solution of fractional order differential or integral equations, the Mittag-Leffler function with its generalizations appears. It is a function of one parameter defined by the power series using the gamma function and it is a natural extension of the exponential, hyperbolic and trigonometric functions. In the book we define the extended and generalized Mittag-Leffler function and give its properties. For different parameter choices, corresponding known generalizations of the Mittag-Leffler function can be deduced: the Wiman generalization, also known as Mittag-Leffler function of two parameters, Prabhakar’s function, Shukla-Prajapati’s function, Salim-Faraj’s function or recent extension defined by Rahman et al. This extended generalized Mittag-Leffler function with the corresponding fractional integral operator (in real domain) is used to obtain fractional generalizations of different types of inequalities, such as inequalities of the Opial, Polya-Szego, Chebyshev, Minkowsky, Hermite, Hadamard and Fejer (for convex, relative convex, m-convex, (h−m)-convex, harmonically convex function, etc). We provide the bounds of various kinds of fractional integral operators containing our extended generalized Mittag-Leffler function and give estimations of these inequalities for different kinds of convex functions. Results are applied on a certain function to establish recurrence relations among Mittag-Leffler functions.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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