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By applying the way of real and functional analysis and estimating the weight functions, we build some lemmas and deduce some Hilbert-type and Hilbert-Hardy-type integral inequalities with the best possible constant factors. The equivalent forms, the reverses and the operator expressions are all considered. The composition formula of two Hilbert-Hardy-type integral operators and some examples are given.

On a composition of two Hilbert-Hardy-type integral operators and related inequalities