WEIGHTED INTEGRAL INEQUALITIES FOR MODIFIED INTEGRAL HARDY OPERATORS

In this article, we study the weak and extra-weak type integral inequalities for the modified integral Hardy operators. We provide suitable conditions on the weights ω, ρ, φ and ψ to hold the following weak type modular inequality $${\mathcal{U}}^{-1}\({\int_{{\mid}{\mathcal{I}}f{\mid}>{\gamma}}}\;{\mathcal{U}}({\gamma}{\omega}){\rho}\){\leq}{\mathcal{V}}^{-1}\({\int}_{0}^{\infty}{\mathcal{V}}(C{\mid}f{\mid}{\phi}){\psi}\),$$ where ${\mathcal{I}}$ is the modified integral Hardy operators. We also obtain a necesary and sufficient condition for the following extra-weak type integral inequality $${\omega}\(\{{\left|{\mathcal{I}}f\right|}>{\gamma}\}\){\leq}{\mathcal{U}}{\circ}{\mathcal{V}}^{-1}\({\int}_{0}^{\infty}{\mathcal{V}}\(\frac{C{\mid}f{\mid}{\phi}}{{\gamma}}\){\psi}\).$$ Further, we discuss the above two inequalities for the conjugate of the modified integral Hardy operators. It will extend the existing results for the Hardy operator and its integral version.