Mapping of Inverse functions. 
Let $\mathsf{f(x)}$ be a differential and an invertible function, such that $\mathsf{f''(x)>0}$ and $\mathsf{f'(x)>0}$.
Prove that$$\mathsf{ f^{-1}\left(\frac{x_1 + x_2 +x_3}{3}  \right) > \frac{f^{-1}(x_1)+f^{-1}(x_2)+f^{-1}(x_3)}{3}}$$

I have no clue, how to start it. I think a graphical solution can be obtained but I am confused about the graph of the inverse function.

This should help : 'Inverse of Convex Strictly Monotone Function'.
Use the definition of concavity with $\alpha=1/3, \beta=2/3$ and so on...
