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 are proved. Also, some finite and infinite decompositions of some type Hardy-Hilbert’s integral operators are given. Indeed, for a non-negative kernel K, two Kernels K1 and K2 are given such that TK = TK1 + TK2 and TK = TK1  + TK2  and also, Tk1 = cTk2 for every constant c. So, the space of bounded linear operators is not strictly convex. Also, as an application of infinite decomposition of some Hardy-Hilbert’s integral operators, the convergence of some series of hypergeometric functions are given.