Abstract: Using the weight coefficient method and real analysis techniques, the author considers the  constructing semi-discrete Hilbert-type reverse inequality with  non-homogeneous kernel. It provides sufficient necessary conditions for constructing such inequality and calculation formulas for the best constant factor. Finally, the operator expression for the inequalities and some special cases are presented.

Key words: semi-discrete Hilbert-type reverse inequality, non-homogeneous kernel, construction theorem, the best constant factor, integral operator, discrete operator