An Elementary Inequality by Non-elementary Means

Problem

An Elementary Inequality by Non-elementary Means

Proof

By the Cauchy-Schwarz inequality,

$\displaystyle\left(\int_0^ax\cos xdx\right)^2\le\left(\int_0^ax^2dx\right)\left(\int_0^a\cos^2xdx\right).$

We continue by evaluating integrals:

$\displaystyle\left(x\sin x\bigg|^a_0-\int_0^a\sin xdx\right)^2\le\frac{a^3}{3}\int_0^a\frac{1+\cos 2x}{2}dx.$

$\displaystyle\left(a\sin a+\cos a-\cos 0\right)^2\le\frac{a^3}{6}\left(x\bigg|_0^a+\frac{1}{2}\sin 2x\bigg|_0^a\right).$

$\displaystyle 6\left(a\sin a+\cos a-1\right)^2\le a^3\left(a+\frac{1}{2}\sin 2a\right).$

$\displaystyle 12\left(a\sin a+\cos a-1\right)^2\le 2a^4+a^3\sin 2a.$

Acknowledgment

Dan Sitaru has kindly communicated in a private message the above problem and its solution.

There is little doubt that the expression $a\sin a+\cos a\;$ betrays the integral origins of the problem. However, the inequality itself is quite elementary looking which makes one curious whether it has a more elementary solution that does not invoke calculus.

 

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