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 .

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

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