Stronger than Jensen's inequality

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 .

• Couldn't both $f'(x),f'(y)=0?$ – zhw. Jul 5 at 16:00