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 &#x03C9;, &#x03C1;, &#x03C6; and &#x03C8; to hold the following weak type modular inequality <TEX>$${mathcal{U}}^{-1}({int_{{mid}{mathcal{I}}f{mid}>{gamma}}};{mathcal{U}}({gamma}{omega}){ ho}){leq}{mathcal{V}}^{-1}({int}_{0}^{infty}{mathcal{V}}(C{mid}f{mid}{phi}){psi}),$$</TEX> where <TEX>${mathcal{I}}$</TEX> is the modified integral Hardy operators. We also obtain a necesary and sufficient condition for the following extra-weak type integral inequality <TEX>$${omega}({{left|{mathcal{I}}f ight|}>{gamma}}){leq}{mathcal{U}}{circ}{mathcal{V}}^{-1}({int}_{0}^{infty}{mathcal{V}}(frac{C{mid}f{mid}{phi}}{{gamma}}){psi}).$$</TEX> 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.