This question is arised when I study the content in the following question entitled by
" lower bound of a special type of convex functions " in here.
Let $f: {\bf R}^n \rightarrow {\bf R}$ be a function in $C^1$ with
$f(sx+(1-s)y)< s f(x) + (1-s) f(y)$, $s\in (0,1)$.
Then $f(y) > f(x) + \nabla f(x) \cdot (y-x)$.
This inequality is followed from the intuition. But I want to know the rigorous proof. Thank you in advance.