The conditions are investigated, under which for all Lebesgue measurable functions f(x) greater than or equal 0 on a semi-axis R+:=(0, infinity ) with a constant C greater than or equal 0 independent of f, satisfied is inequality: {0 integral infinity [Kf(x)]qν(x)dx}1/q [less-than or equal to] C{0 integral infinity [f(x)]pu(x)dx}1/p (1) with measurable weighted functions u(x) greater than or equal 0 and ν(x) greater than or equal 0 and integral operator Kf(x):=0 integral infinity k(x,y)f(y)dy, where measurable in R+×R+ kernel k(x,y) greater than or equal 0 is monotone in one or two variables. Such operators can be exemplified with Laplace, Hilbert transforms etc. Further, the comparison theorems for (1)-type inequalities with the similar inequalities on a cone of non-growing functions for certain-type Volterra operators are proved.