We continue our previous work on a problem of Janiec connected with a uniqueness theorem, of Cartan-Gutzmer type, for holomorphic mappings in ℂⁿ. To solve this problem we apply properties of (j;k)-symmetrical functions.

We give a new proof of a Phragmén Lindelöf theorem due to W.H.J. Fuchs and valid for an arbitrary open subset $U$ of the complex plane: if $f$ is analytic on $U$, bounded near the boundary of $U$, and the growth of $j$ is at most polynomial then either $f$ is bounded or $U\supset \left\{\right|z|\>r\}$ for some positive $r$ and $f$ has a simple pole.

In this paper we give some analytic formulas for the hyperbolic (Harnack) distance between two contractions which permit concrete computations in several situations, including the finite-dimensional case. The main consequence of these formulas is the proof of the Schwarz-Pick Lemma. It modifies those given in [13] by the avoidance of a general Schur type formula for contractive analytic functions, more exactly by reducing the case to the more manageable situation when the function takes as values...

Let $𝒜$ denote the set of functions $F$ holomorphic in the unit disc, normalized clasically: $F\left(0\right)=0,F^{\text{'}}\left(0\right)=1$, whereas $A\subset 𝒜$ is an arbitrarily fixed subset. In this paper various properties of the classes $A_{\alpha },\alpha \in C\{-1,-\frac{1}{2},...\}$, of functions of the form $f=F*k_{\alpha }$ are studied, where $F\in .A$, $k_{\alpha }\left(z\right)=k(z,\alpha )=z+\frac{1}{1+\alpha }{z}^2+...+\frac{1}{1+(n-1)\alpha }{z}^n+...$, and $F*k_{\alpha }$ denotes the Hadamard product of the functions $F$ and $k_{\alpha }$. Some special cases of the set $A$ were considered by other authors (see, for example, [15],[6],[3]).

By using the theory of first-order differential subordination for functions with fixed initial coefficient, several well-known results for subclasses of univalent functions are improved by restricting the functions to have fixed second coefficient. The influence of the second coefficient of univalent functions becomes evident in the results obtained.