Аннотация:
Let $\mathcal{A}$ denote the set of all analytic functions $f$ on the unit disk normalised so that $f(0)=f'(0)-1=0$ and $\mathcal{S} $ denote the subclass of functions $f \in \mathcal{A}$ which are univalent in the unit disk.
The problem of estimating sharp bound for successive coefficients, namely, $\big | |a_{n+1}|-|a_n| \big | $, is an
interesting necessary condition for a function to be in $\mathcal{S}$. This problem was first studied by Goluzin [1]
with an idea to solve the Bieberbach conjecture. Hayman [3]
proved in 1963 that
\begin{equation}\label{deq5}
\big | |a_{n+1}|-|a_n| \big | \leq A, \quad n=1,2,3,\dots,
\end{equation}
where $A\geq 1$ is an absolute constant, for $f \in \mathcal{S}$ with the form $f(z)=z+\sum_{n=2}^{\infty} a_n z^n$. It is still an open problem to find the minimal value of $A$
which works for all $f \in \mathcal{S}$, however, the best known bound as of now is $3.61$ which is due to
Grinspan [2]. In $1978$, Leung [4] proved that $A=1$ for starlike functions implies the Bieberbach conjecture, so it is interesting to find the minimal value of $A$ for certain other subfamilies of univalent functions.
We are considering some subclasses of the class of univalent functions, mainly, the family $\mathcal{S}_{\gamma}(\alpha)$ of $\gamma$-spirallike functions of order $\alpha$ which is defined as
$$
\mathcal{S}_\gamma (\alpha) = \left \{f\in {\mathcal A}: \, {\rm Re} \left ( e^{-i\gamma}\frac{zf'(z)}{f(z)}\right )>\alpha \cos \gamma\,
, z\in \mathbb{D}\right\},
$$
where $ \alpha \in [0,1)$ and $\gamma\in (-\pi/2, \pi/2)$. Note that $\mathcal{S}_0 (\alpha)=:\mathcal{S}^*{(\alpha)}$ is the usual class of starlike functions of order $\alpha$, and $\mathcal{S}^*(0)=\mathcal{S}^*$. A function $f \in \mathcal{A}$ is called convex of order $\alpha$ if and only if $zf'(z)$ belongs to $\mathcal{S}^*{(\alpha)}$ for some $\alpha \in [ 0,1)$. The class of all convex functions of order $\alpha$ is denoted by $\mathcal{C}(\alpha)$.
Our objective is to obtain results related to successive coefficients for the classes $\mathcal{S}^*{(\alpha)}$, $\mathcal{C}(\alpha)$, $\mathcal{S}_{\gamma}(\alpha)$ and other related classes of functions.
Язык доклада: английский
Список литературы
G. Golusin, “On distortion theorems and coefficients of univalent functions”, Rec. Math. [Mat. Sbornik] N.S., 19(61) (1946), 183–202
A. Z. Grin\vspan, “The sharpening of the difference of the moduli of adjacent coefficients of schlicht functions”, Some Problems in Modern Function Theory (Proc. Conf. Modern Problems of Geometric Theory of Functions, Inst. Math., Acad. Sci. {Ussr}, Novosibirsk, 1976) (Russian), 1976, 41–45
W. K. Hayman, “On successive coefficients of univalent functions”, J. London Math. Soc., 38 (1963), 228–243
Yuk Leung, “Successive coefficients of starlike functions”, Bull. London Math. Soc., 10:2 (1978), 193–196