Ngô Quốc Anh

May 10, 2008

An integral inequality

Filed under: Các Bài Tập Nhỏ, Giải Tích 2 — Ngô Quốc Anh @ 3:48

Let [0,1]\rightarrow \mathbb{R}^{ + } be a continuous function and let be a natural number. Prove that

\int\limits_{0}^{1}f^{n}(x^{n})dx\geq \frac {1}{n}\left(\frac {n +1}{n }\right)^{n - 1}\left(\int\limits_{0}^{1}f(x)dx\right)^{n}.

Solution. Recall Hôlder inequality

, , \frac {1}{a} + \frac {1}{b} = 1,

|\int_{K}h.g|\leq(\int_{K}|h|^{a})^{1/a}(\int_{K}|g|^{b})^{1/b}.

Then

\int_{0}^{1}f(x)dx = \int_{0}^{1}y^{\frac {n - 1}{n^2}}(f(y)y^{\frac {1 - n}{n^2}})dy\leq (\int_{0}^{1}y^{\frac {1}{n}}dy)^{\frac {n - 1}{n}}(\int_{0}^{1}f^{n}(y)y^{\frac {1}{n} - 1}dy)^{\frac {1}{n}}

taking the n-power and in the second integral at RHS , do change variable will give the inequality.

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