For a convex function $f: X \to \mathbf{R}$, there is the famous Jensen's inequality
$$f\left(\frac{\sum_{i=1}^n x_i}{n}\right) \leq \frac{\sum_{i=1}^n f(x_i)}{n}$$
Is there a lower bound to $f\left(\frac{\sum_{i=1}^n x_i}{n}\right)$ ? $f$ is a differentiable too in my case.
There is no inequality of the type $$f\left(\frac{\sum\limits_{k=1}^{n} x_i} {n}\right) \geq C \frac { \sum\limits_{k=1}^{n} f(x_i)} n$$ for convex functions.
For example $f(\frac {a +b} 2) \geq C \frac {f(a)+f(b)} 2$ fails for the convex function $f(x)=e^{-x}$. Take $a=0$ and check that the inequality cannot hold for all $b \in \mathbb R$.