Foundations of generalized Prabhakar-Hilfer fractional calculus with applications

Here we introduce the generalized Prabhakar fractional calculus and we also combine it with the generalized Hilfer cal-culus. We prove that the generalized left and right side Prab-hakar fractional integrals preserve continuity and we find tight upper bounds for them. We present several left and right side generalized Prabhakar fractional inequalities of Hardy, Opial and Hilbert-Pachpatte types. We apply these in the setting of generalized Hilfer calculus