We present a detailed great variety of Hardy type fractional inequalities under convexity and Lp norm in the setting of generalized Prabhakar and Hilfer fractional calculi of left and right integrals and derivatives. The radial multivariate case of the above over a spherical shell is developed in detail to all directions. Many inequalities are of vectorial splitting rational Lp type or of separating rational Lp type, others involve ratios of functions and of fractional integral operators.
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