Bounding a function times the mean of the reciprocal

Question

My question might be trivial. Can one characterize all the periodic regular functions $f:[-\pi,\pi]\rightarrow\mathbb{R}$, say for symplicity $f\geq1$, such that $$f(x)\int_{-\pi}^{\pi}\frac{1}{f(y)}dy\leq 2\pi?$$

Answer 1

$f$ must be a constant function. Namely, the regular positive function $\varphi(x) := 1/f(x)$ must satisfy $$ \varphi(x) \geq \frac{1}{2\pi}\int_{-\pi}^\pi \varphi(y)\, dy \qquad \forall x\in [-\pi, \pi], $$ i.e. $\varphi$ must be $\geq$ its integral mean at every point. But this can happen if and only if $\varphi$ is constant.