In this paper, we aim to determine some results of the generalized Bessel–Maitland function in the field of fractional calculus. Here, some relations of the generalized Bessel–Maitland functions and the Mittag-Leffler functions are considered. We develop Saigo and Riemann–Liouville fractional integral operators by using the generalized Bessel–Maitland function, and results can be seen in the form of Fox–Wright functions. We establish a new operator Zν,η,ρ,γ,w,a+μ,ξ,m,σϕ and its inverse operator Dν,η,ρ,γ,w,a+μ,ξ,m,σϕ, involving the generalized Bessel–Maitland function as its kernel, and also discuss its convergence and boundedness. Moreover, the Riemann–Liouville operator and the integral transform (Laplace) of the new operator have been developed.
In this article, we established a new version of generalized fractional Hadamard and Fejér–Hadamard type integral inequalities. A fractional integral operator (FIO) with a non-singular function (multi-index Bessel function) as its kernel and monotone increasing functions is utilized to obtain the new version of such fractional inequalities. Our derived results are a generalized form of several proven inequalities already existing in the literature. The proven inequalities are useful for studying the stability and control of corresponding fractional dynamic equations.
In this paper, we discuss the multi-index Mittag Leffler k-function and Bessel k-function of the first kind in fractional calculus. We investigate fractional integral operators (Saigo's, Erdelyi Kober, Reimann Liouvill, Weyl) and extend with the product of multi-index Mittag Leffler k-function to the Bessel k-function of the first kind. Here, we establish new theorems that provide the image of multi-index Mittag Leffler and Bessel k-functions under these k-fractional operators. These results are derived in general behave and give several well-known results in the theory of multi-index k-functions.
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