Coefficient estimates for negative powers of the derivative of univalent functions

Letf be a one-to-one analytic function in the unit disc withf′(0)=1. We prove sharp estimates for certain Taylor coefficients of the functions(f′) ^p , wherep<0. The proof is similar to de Branges’ proof of Bieberbach’s conjecture, and thus relies on Löwner’s equation. A special case leads to a generalization of the usual estimate for the Schwarzian derivative off. We use this to improve known estimates for integral means of the functions |f′| ^p for integersp??2.     

Typically real mean univalent functions of large growth

Typically real normalized κ-dimensionally mean univalent functionsf on |z|<1 are considered for which $\mathop {\lim \sup }\limits_{r \uparrow 1} \{ (1 - r)^2 \mathop {\max }\limits_{0 \leqslant \theta \leqslant 2\pi } \left| {f(re^{i\theta } )} \right|\} > 0$ . Lets=logf(z) forz in the unit disc cut along (?1, 0]. A theorem is proved concerning the area of the Riemann surface over thes-plane which distinguishes the two cases ?1≦κ<+∞ and κ=+∞.     

On the mutual growth of neighboring coefficients of univalent functions

In the class S of functionsf(z)=z+c₂z²+c₃z₃+..., regular and univalent in ¦z¦<1, the following bound is obtained: c_n+1¦–¦c_n<4.26, n=1, 2, ...Translated from Matematicheskie Zametki, Vol. 4, No. 6, pp. 715–722, December, 1968.I wish to express my sincere thanks to I. M. Milin for a series of useful conversations.     

On the zeros of univalent functions with univalent derivatives

Summary A family, E, consisting of normalised univalent functions with univalent derivatives is studied with regard to the zeros of these functions. Entrata in Redazione il 29 marzo 1978.     

Series of univalent functions

We are considering a class S of functions F(z), F(0) = 0, F′(0) = 1 that are univalent and regular in the circle ¦z¦ < 1, and its subclasses s _h ^* and K of starlike functions of order h and of convex functions respectively. Among others, we establish the following results: If F(z)εs and 0 < α < 1, then IfF (z) ε s (0 < a < 1) and $$\begin{gathered} 1 + \operatorname{Re} {{z_1 F^n \left( {z_1 } \right)} \mathord{\left/ {\vphantom {{z_1 F^n \left( {z_1 } \right)} {F'\left( {z_1 } \right)}}} \right. \kern-\nulldelimiterspace} {F'\left( {z_1 } \right)}} = \operatorname{Re} {{\alpha z_1 F''\left( {\alpha z_1 } \right)} \mathord{\left/ {\vphantom {{\alpha z_1 F''\left( {\alpha z_1 } \right)} {F'\left( {\alpha z_1 } \right)}}} \right. \kern-\nulldelimiterspace} {F'\left( {\alpha z_1 } \right)}} \hfill \\ \left( {2 - \sqrt 3< \left| {z_1 } \right| = r< 1} \right) \hfill \\ \end{gathered} $$ then we obtain the domain of values of the point αz₁.     

Almost locally univalent functions

We study ALU holomorphic functions defined in the introduction, and prove two criteria for a function holomorphic in the unit disk to be ALU.     

Radial growth of Bloch functions in the complex ball

We show that N.G. Makarov’s law of the iterated logarithm governs the radial growth of weighted Bloch functions in the complex unit ball. The argument is based on an analog of M. Weiss’ theorem for special lacunary series in the ball.     

Structure formulas for univalent functions

With the aid of special integrating factors for the Loewner-Kufarev equation, one determines the general form of the structure formulas for functions of the class S and one obtains a series of new concrete structure formulas for certain subclasses of functions from S.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 125, pp. 128–134, 1983.