Integral inequality

Real Analysis
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Grigorios Kostakos
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Integral inequality

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Post by Grigorios Kostakos »

Let $f,g:[a,b]\longrightarrow\mathbb{R}$ be continuous functions and $p\in(1,2)$. Prove that $$\displaystyle\bigg(\int_a^b\big|f(x)+g(x)\big|\,^p\,dx\bigg)^{\frac{2}{p}}+(p-1)\bigg(\int_a^b\big|f(x)-g(x)\big|\,^p\,dx\bigg)^{\frac{2}{p}}\leqslant 2\,\bigg(\int_a^b\big|f(x)\big|\,^p\,dx\bigg)^{\frac{2}{p}}+2\,\bigg(\int_a^b\big|g(x)\big|\,^p\,dx\bigg)^{\frac{2}{p}}.$$
Grigorios Kostakos
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