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#### lfdahl

##### Well-known member

- Nov 26, 2013

- 719

Prove the inequality:

$$\int_{0}^{1}\frac{f(x)}{f(x+\frac{1}{2})} \,dx \geq 1.$$

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- Nov 26, 2013

- 719

Prove the inequality:

$$\int_{0}^{1}\frac{f(x)}{f(x+\frac{1}{2})} \,dx \geq 1.$$

- Apr 22, 2018

- 251

$$\left(\int_0^1f(x)dx\right)^2\ \le\ \left(\int_0^1\frac{f(x)}{f\left(x+\frac12\right)}dx\right)^2\left(\int_0^1f\left(x+\frac12\right)dx\right)^2.$$

But $f(x)$ has period $1$ and so $\displaystyle\int_0^1f\left(x+\frac12\right)dx=\int_0^1f(x)dx$. The result follows.

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- Nov 26, 2013

- 719

An alternative solution:

\geq \int_{0}^{\frac{1}{2}} 2\sqrt{\frac{f(x)}{f(x+\frac{1}{2})}\cdot \frac{f(x+\frac{1}{2})}{f(x)} }dx \\=2\int_{0}^{\frac{1}{2}}dx = 1.\]

- Jun 29, 2017

- 79

- Apr 22, 2018

- 251

$$X+Y\ \ge\ 2\sqrt{XY}.$$

This comes from the AM–GM inequality, or alternatively from $\left(\sqrt X-\sqrt Y\right)^2\ge0$.

- Jun 29, 2017

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