We state and prove some new refined Hardy type inequalities using the notation of superquadratic and subquadratic functions with an integral operator A k defined by
We state, prove, and discuss new general inequality for convex and increasing functions. As a special case of that general result, we obtain new fractional inequalities involving fractional integrals and derivatives of Riemann-Liouville type. Consequently, we get the inequality of H. G. Hardy from 1918. We also obtain new results involving fractional derivatives of Canavati and Caputo types as well as fractional integrals of a function with respect to another function. Finally, we apply our main result to multidimensional settings to obtain new results involving mixed Riemann-Liouville fractional integrals.
Abstract. We state and prove some new weighted Hardy type inequalities with an integral operator A k defined bywhere k : Ω 1 × Ω 2 → R is a general nonnegative kernel, (Ω 1 , μ 1 ) and (Ω 2 , μ 2 ) are measure spaces andIn particular, the obtained results unify and generalize most of the results of this type (including the classical ones by Hardy, Hilbert and Godunova).
Mathematics subject classification (2000): 26D15, 26D30, 26D32.Keywords and phrases: the Hardy inequality, the Hilbert inequality, Inequalities, Hardy type inequalities, convex function, kernel, the Hardy operator with general kernel.
R E F E R E N C E
Abstract. The main goal of the paper is to state and prove the new general inequality for convex and increasing functions. We introduce some new inequalities by involving some fractional integrals and fractional derivatives of Riemman-Liouville, Canavati, Hadamard and Erdelyi-Kóber type and apply our result to multidimensional setting to obtain new results involving mixed Riemman-Liouville fractional integrals.Mathematics subject classification (2010): Primary 26D10, Secondary 26D15.
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