I'm interested by the following problem :
Let $f(x)$ be a twice differentiable function on an interval $I$ with :
1)$f''(x)\geq 0\quad \forall x \in I$
2)$f(x)\neq \text{constant function} $
Then we have for $x,y \in I$: $$f\Big(\frac{x+y}{2}\Big)\leq f\Big(\frac{1}{2}\Big(\frac{xf'(x)+yf'(y)}{f'(x)+f'(y)}\Big)+\frac{x+y}{4}\Big)\leq \frac{f(x)+f(y)}{2}$$
I have tried a lot of things like Karmata's inequality but it fails always ...I'm a little bit desperate with this .
Maybe majorization is a good way but the theorems I use are too weak .
Any hints are appreciable .
Thanks a lot for your time .