Abstract. Let $\omega(x)$ and $v(x)$ be non-negative functions on $[0,\infty )$. We find the best constant in the integral inequality $$\left(\int_0^\infty[Tg(x)]^q\omega(x)dx\right )^{1/q} \le C\left(\int_0^\infty g^p(x)v(x)dx\right )^{1/p},$$ in the class of non-negative, non-increasing or non-decreasing functions $g$ when $0< p\le q< \infty $, $0< p\le1$, and $T$ is a general integral operator of the form $$Tg(x)=\int_0^\infty k(x,y)g(y)dy, k(x,y)\ge0.$$ The similar problem is solved for the reversed inequality provided that $1\le p\le q< \infty $. A number of applications are given. In particular, some recently obtained sharp reversed Hardy type inequalities are analyzed in this connection.

AMS Subject Classification (1991): 26D10

Received June 13, 1994. (Registered under 5611/2009.)