The analogous version of Grüss inequalities has been established using the generalized hypergeometric function fractional integral operators. The results are generalizations of Grüss type inequalities in fractional integral operators. Our main deduction will break into results noted for appropriate changes of fractional integral parameter and degree of fractional operator.
Mittag-Leffler functions has many applications in various areas of Physical, biological ,applied, earth Sciences and Engineering. It is used in solving problems of fractional order differential, integral and difference equations. In this paper, we aim to define the m-parameter Mittag-Leffler function, which can be reduced to various already known extensions of Mittag-Leffler function. We then, discuss its various properties like recurrence relations, differentiation formula and integral representations. We also represent the new m-parameter Mittag-Leffler function in terms of some known special functions such as Generalized hypergeometric function, Mellin Barnes integral, Wright hypergeometric function and Fox H-function. We also discuss its various integral transforms like Euler-Beta, Whittaker, Laplace and Mellin transforms. Further, fractional differential and integral operators are considered to discuss few properties of m-parameter Mittag-Leffler function. Also, we use the m-parameter Mittag-Leffler function to define a generalization of Prabhakar integral and discuss its properties. Further, relation of m-parameter Mittag Leffler function with various other functions such as exponential, trigonometric, hypergeometric and algebraic functions is obtained and represented graphically using MATHEMATICA 12.
The Object of the present paper is to establish some interested theorems on Euler type integral involving extended Mittag-Leffler function. Further, we reduce some special cases involving various known functions like Wiman function, Prabhakar function, exponential and Binomial functions.
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