Integral analog of one generalization of the Hardy inequality and its applications - Mulyava, O. M. (Ukrains’kyi Matematychnyi Zhurnal, 2006)
Under certain conditions on continuous functions $μ, λ, a$, and $f$, we prove the inequality$$\int\limits_0^y {μ (x)λ (x)f\left( {\frac{{\int_0^x {λ (t)a(t)dt} }}{{\int_0^x {λ (t)dt} }}} \right)dx \leqslant K\int\limits_0^y {μ (x)λ (x)f(a(x))} dx,} y \leqslant ∞ ,$$and describe its application to the investigation of the problem of finding conditions under which Laplace integrals belong to a class of convergence.