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Prove this integral inequality

Source: 2009 Jozsef Wildt International Mathematical Competition

April 26, 2020
integrationinequalitiescalculus

Problem Statement

If the function f:[0,1]→(0.+∞)f:[0,1]\to (0.+\infty) is increasing and continuous, then for every a≥0a\geq 0 the following inequality holds: ∫01xa+1f(x)dx≤a+1a+2∫01xaf(x)dx\int \limits_0^1 \frac{x^{a+1}}{f(x)}dx \leq \frac{a+1}{a+2} \int \limits_0^1 \frac{x^{a}}{f(x)}dx
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