On univalent functions with two preassigned values

Abstract:

Let ${\mathfrak {M}_M}$ denote the class of functions analytic and univalent in the unit disc $\Delta$ subject to the conditions \[ f(0) = 0,\quad f({z_0}) = {z_0},\quad |f(z)| < M,\] where ${z_0},{z_0} \ne 0$, is a fixed point of $\Delta$ and $1 \leqq M \leqq \infty$. In the present note, we determine by the method of circular symmetrization, the exact value of the “Koebe constant” for the class ${\mathfrak {M}_M}$. We also determine Koebe sets for the class $\mathfrak {M}_M^ \ast$ consisting of the starlike functions, and for $\mathfrak {M}_M^\alpha$, consisting of all functions mapping $\Delta$ onto domains convex in the direction ${e^{i\alpha }}$. By “Koebe set” we understand the set $\mathcal {K}({\mathfrak {M}_M}),\mathcal {K}({\mathfrak {M}_M})$.