Abstract:

Choosing $ a, b$ as the matching parameters, we can sue the weight function method to obtain Hilbert-type series inequality $\sum\limits_{m=1}^{\infty}\sum\limits_{n=1}^{\infty}K(m, n)a_{m}b_{n}\leq M(a, b)\|\tilde{a}\|_{p, \alpha(a, b)}\|\tilde{b}\|_{q, \beta(a, b)}.$ in the paper, the problem of how to choose $a, b $ in order to make $ M(a, b)$ the best constant factor in inequality with quasi-homogeneous kernels are discussed, necessary and sufficient conditions are obtained for $a, b $ to the best matching parameters, the formula for the best constant factor is obtained. Finally, their applications to solving operator morn are discussed.

Key words: Hilbert-type series inequality, Quasi-homogeneous kernel, The best constant factor, The best matching parameter, Operator norm