Coefficient inequalities for conformal mappings with homeomorphic continuation
The author proves several coefficient inequalities for functions in the class S, for Bieberbach-Eilenberg functions (i.e., functions analytic in the unit disk and with \(f(z_ 1)f(z_ 2)\neq 1\) for \(| z_ 1| <1\), \(| z_ 2| <1)\), and for Gelfer functions (i.e., functions of the form \(\phi (z)=1+\sum^{\infty}_{n=1}b_ nz^ n\) with \(\phi (z_ 1)+\phi (z_ 2)\neq 0\) for \(| z_ 1| <1\), \(| z_ 2| <1)\). Most of the theorems involve functions with homeomorphic extensions. An example of one of the shorter theorems has to suffice: Let \(f(z)=z+c_ 2z^ 2+...\in S,\) and suppose q(w) is analytic on the set \(d=C[f(| z| <1)\}.\) The coefficient \(a_ n\) \((n=\pm 1,\pm 2,...)\) are determined from the expansion \(q(f(z))=\sum^{\infty}_{n=- \infty}a_ nz^ n,\) which is valid in some annulus \(r<| z| <1\). Then the inequality \[ \sum^{\infty}_{n=1}n| a_ n|^ 2\leq K^ 2_ s(q/f)\sum^{\infty}_{n=1}n| a_{-n}|^ 2 \] holds. A necessary and sufficient condition for equality is also given.
pairs of functions with nonoverlapping values
scientific article; zbMATH DE number 3908782