关键词: integral, inequality, continuous function

Abstract: Given two positive constantsαandβ,we prove that the integral inequality∫₀¹f^α+β(x)dx≥∫₀¹ f^α(x)x^βdx holds for all non-negative valued continuous functions f satisfying∫_x¹f(t)dt≥∫_x¹tdt for x∈[0,1] if and only if α+β≥1. This solves an open problem proposed recently by Ngo, Thang, Dat, and Tuan.

Key words: integral, inequality, continuous function