Further inequalities for the generalized k-g-fractional integrals of functions with bounded variation

Authors : Sever DRAGOMİR

Pages : 49-72

Doi:10.31801/cfsuasmas.542665

View : 41 | Download : 10

Publication Date : 2020-06-30

Article Type : Research Paper

Abstract :Let g be a strictly increasing function on insert ignore into journalissuearticles values(a,b);, having a continuous derivative g′ on insert ignore into journalissuearticles values(a,b);. For the Lebesgue integrable function f:insert ignore into journalissuearticles values(a,b);→C, we define the k-g-left-sided fractional integral of f by S_{k,g,a+}finsert ignore into journalissuearticles values(x);=∫_{a}^{x}kinsert ignore into journalissuearticles values(ginsert ignore into journalissuearticles values(x);-ginsert ignore into journalissuearticles values(t););g′insert ignore into journalissuearticles values(t);finsert ignore into journalissuearticles values(t);dt, x∈insert ignore into journalissuearticles values(a,b] and the k-g-right-sided fractional integral of f by S_{k,g,b-}finsert ignore into journalissuearticles values(x);=∫_{x}^{b}kinsert ignore into journalissuearticles values(ginsert ignore into journalissuearticles values(t);-ginsert ignore into journalissuearticles values(x););g′insert ignore into journalissuearticles values(t);finsert ignore into journalissuearticles values(t);dt, x∈[a,b);, where the kernel k is defined either on insert ignore into journalissuearticles values(0,∞); or on [0,∞); with complex values and integrable on any finite subinterval. In this paper we establish some new inequalities for the k-g-fractional integrals of functions of bounded variation.Examples for the generalized left- and right-sided Riemann-Liouville fractional integrals of a function f with respect to another function g and a general exponential fractional integral are also provided.
Keywords : Generalized Riemann Liouville fractional integrals, Hadamard fractional integrals, Functions of bounded variation, Ostrowski type inequalities

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