On Finite and Infinite Decomposition of Some Hilbert’s Type Inequalities

In this work, some hardy-hilbert's integral inequalities with the best possible constants is proved. also, some finite and infinite decompositions of sometype hardy-hilbert's integral operators is given. indeed, for a non-negative kernel k, two kernels k1 and k2 is given such that tk = tk1 + tk2 and ∥tk∥ =∥tk1∥ + ∥tk2∥. so, the space of bounded linear operators is strictly convex.also, as an application of infinite decomposition of some hardy-hilbert's integraloperators, the convergence of some series of hypergeometric functions is given.