Question 5
Let $\mathbb R_{\gt 0}$ be the set of positive real numbers. Determine all functions $f:\mathbb R_{\gt 0}\to\mathbb R_{\gt 0}$ such that
$$\sqrt{\frac{x^2+f(y)^2}{2}}\ge\frac{f(x)+y}{2}\ge\sqrt{xf(y)}$$
for every $x,y\in\mathbb R_{\gt 0}$.
$$\sqrt{\frac{x^2+f(y)^2}{2}}\ge\frac{f(x)+y}{2}\ge\sqrt{xf(y)}$$
for every $x,y\in\mathbb R_{\gt 0}$.