On some new Hardy-type inequalities

In this paper, we give some new types of the classical Hardy integral inequality by including a second parameter q and using weighted mean operators S-1 := (S-1)(g)(w) and S-2 := (S-2)(g)(w) defined by S-1(x) = 1/W(x) integral(x)(a) w(t)g(f(t))dt, S-2(x) = integral(x)(a) w(t)/W(t)g(f(t))dt, with W(x) = integral(x)(0) w(t)dt, for x is an element of(0,+infinity), where w is a weight function and g is a real continuous function on (0,+infinity).