On the sum of powered distances to certain sets of points on the circle

In this paper we consider an extremal problem in geometry. Let $lambda$ be a real number and $A$, $B$ and $C$ be arbitrary points on the unit circle $Gamma$. We give full characterization of the extremal behavior of the function $f(M,lambda)=MA^lambda+MB^lambda+MC^lambda$, where $M$ is a point on the unit circle as well. We also investigate the extremal behavior of $sum_{i=1}^nXP_i$, where $P_i, i=1,...,n$ are the vertices of a regular $n$-gon and $X$ is a point on $Gamma$, concentric to the circle circumscribed around $P_1...P_n$. We use elementary analytic and purely geometric methods in the proof.