Convex functions exercises I am trying to solve the two exercizes:


*

*Given $f:(0,\infty) \rightarrow \mathbb{R}$ is a convex function, and $\displaystyle\lim_{x \rightarrow \infty} f(x) = 0$, prove that, $g(x) = \frac{f(x)}{x}$ is an increasing monotone function in $(0,\infty)$.

*Prove the following inequality for each $x,y > 0$: 
$x\ln x + y\ln y \geq (x+y)\ln(\frac{x+y}{2})$
Thank you!

Hint
$2.\;\;$ $f$ is concave function if for $p,q\in[0,1]$ we have
$$f(px+qy)\ge pf(x)+qf(y)$$
take $p=\frac{x}{x+y}$ and $q=\frac{y}{x+y}$ and $f=\ln$
Added This is the  full answer:
We have
$$\ln\left(\frac{2}{x+y}\right)=\ln\left(\frac{x}{x+y}\frac 1 x+\frac{y}{x+y}\frac 1 y\right)\ge\frac{x}{x+y}\ln\left(\frac 1 x\right)+\frac{y}{x+y}\ln\left(\frac 1 y\right)$$
now multiply this inequality by $-(x+y)$ and you have your result.
