By the use of techniques of real analysis and weight functions, we obtain two lemmas and build a few equivalent conditions of a Hardy-type integral inequality with a non-homogeneous kernel, related to a parameter where the constant factor is expressed in terms of the extended Riemann zeta function. Meanwhile, a few equivalent conditions for two kinds of Hardy-type integral inequalities with the homogeneous kernel are deduced. We also consider the operator expressions.
"Equivalent conditions of a Hardy-type integral inequality related to the extended Riemann zeta function." Adv. Oper. Theory 2 (3) 237 - 256, Summer 2017. https://doi.org/10.22034/aot.1703-1132