Question Number 161162 by metamorfose last updated on 13/Dec/21
![f :]0,+∞[→]0,+∞[ is convex function for n≥2 an integer , prove : (f(1)^(f(1)) f(2)^(f(2)) ...f(n)^(f(n)) )^(1/(f(1)+f(2)+...+f(n))) +(f(1)f(2)...f(n))^(1/n) ≤f(1)+f(n)](Q161162.png)
$$\left.{f}\::\right]\mathrm{0},+\infty\left[\rightarrow\right]\mathrm{0},+\infty\left[\:{is}\:{convex}\:{function}\right. \\ $$$${for}\:{n}\geqslant\mathrm{2}\:{an}\:{integer}\:,\:{prove}\:: \\ $$$$\left({f}\left(\mathrm{1}\right)^{{f}\left(\mathrm{1}\right)} {f}\left(\mathrm{2}\right)^{{f}\left(\mathrm{2}\right)} ...{f}\left({n}\right)^{{f}\left({n}\right)} \right)^{\frac{\mathrm{1}}{{f}\left(\mathrm{1}\right)+{f}\left(\mathrm{2}\right)+...+{f}\left({n}\right)}} +\left({f}\left(\mathrm{1}\right){f}\left(\mathrm{2}\right)...{f}\left({n}\right)\right)^{\frac{\mathrm{1}}{{n}}} \leqslant{f}\left(\mathrm{1}\right)+{f}\left({n}\right) \\ $$