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By the use of the way of Real and Functional Analysis and estimating the weight functions, we build some lemmas and deduce some Hilbert-Hardy-type integral inequalities. The equivalent forms and the reverses are all considered. Two kinds of Hilbert-Hardy-type integral operators are defined and the composition formulas of the operators are given. Mathematics subject classification (2010): 26D10, 26D15, 40A05, 33B15.

Two kinds of compositions of Hilbert-Hardy-type integral operators and the related inequalities