Star-like function

univalent star-like function

A function $w=f(z)$ which is regular and univalent in the disc $|z|<1$, $f(0)=0$, and maps $|z|<1$ onto a star-like domain with respect to $w=0$. A function $f(z)$, $f(z)\neq0$ in $0<|z|<1$, $f(0)=0$, $f'(0)\neq0$, regular in $|z|<1$, is star-like in this disc if and only if it satisfies the condition

$$\operatorname{Re}\left[\frac{zf'(z)}{f(z)}\right]>0,\quad|z|<1.$$

The family of star-like functions in $|z|<1$, normalized so that $f(0)=0$, $f'(0)=1$, forms the class $S^*$, which admits a parametric representation by Stieltjes integrals:

$$f(z)=z\exp\left[-2\int_{-\pi}^{\pi}\log(1-e^{-it}z)d\mu(t)\right],$$

where $\mu(t)$ is a non-decreasing function on $[-\pi,\pi]$, $\mu(\pi)-\mu(-\pi)=1$.

For the class $S^*$ the coefficient problem has been solved; sharp estimates have been found for $|f(z)|$, $|f'(z)|$, $\arg f(z)$, $\arg f'(z)$ (the argument of the function is the branch that vanishes at $z=0$). The extremal functions for these estimates are $f(z)=z/(1-e^{i\theta}z)^2$, where $\theta$ is real. The class $S^*$ of functions $f(z)$ is related to the class of functions $\phi(z)$, $\phi(0)=0$, $\phi'(0)=1$, that are regular and univalent in $|z|<1$ and map $|z|<1$ onto a convex domain, by the formula $z\phi'(z)=f(z)$.

A star-like function such that

$$\operatorname{Re}\left[\frac{zf'(z)}{f(z)}\right]>\alpha,\quad|z|,\alpha<1,$$

is called a star-like function of order $\alpha$ in $|z|<1$.

Attention has also been given to univalent star-like functions in an annulus (see [1]), $p$-valent star-like functions and weakly star-like functions in a disc (see [2], [4]), $\epsilon$-locally star-like functions (see [1]), and functions which are star-like in the direction of the real axis (see [3]). For star-like functions of several complex variables, see [5].

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