Let $f$ be a twice differentiable function on $\mathbb{R}$ with a single inflection point, let $S$ be a fixed real number and let $$g(x) = f(x) + (n-1)f(\frac{S-x}{n-1}).$$ If $x_1, x_2,...,x_n$ are real numbers such that $x_1+x_2+... +x_n=S$, then $$\text{inf } g(x) \leq f(x_1) +f(x_2) +... +f(x_n) \leq \text{sup } g(x) $$
I found it in a book of algebraic inequalities... But the proof was not presented by the author... I tried to prove it, but could not proceed...
Any help on this will be appreciated...