An extension of some integral inequalities

We denote by R, n ≥ 1, the Euclidean space of dimension n endowed by the standard Lebesgue measure dx. Let L(Ω) be the space of real functions defined in Ω ⊂ R such that ∫ Ω f(x) dx <∞. In their paper [1], Bai-Ni Guo and Xin Jiang state the following: Theorem 1. Let Ω be a domain in R and f, g ∈ L(Ω) such that f ≥ 0 and g ≥ 0 and let I(f) = ∫ Ω f(x) dx. Further let h : Ω → R such that h ∈ L(Ω). If