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Angle distortion theorems for starlike and spirallike functions with respect to a boundary point. (English) Zbl 1137.30003

Let ${\Delta }$ be the open unit disk in the complex plane $𝒞$. By $\text{Hol}\left({\Delta },𝒞\right)$ it is denoted the set of all holomorphic functions in ${\Delta }$.

A univalent function $h$ on ${\Delta }$ is called spirallike (resp., starlike) on ${\Delta }$ if for some $\mu \in 𝒞$ with $Re\mu >0$ (resp., $\mu \in ℛ$ with $\mu >0$) and for each $t\ge 0$, the element ${e}^{-\mu t}h\left(z\right)$ belongs to $h\left({\Delta }\right)$ whenever $z\in {\Delta }$.

It is clear that $0\in \overline{h\left({\Delta }\right)}$. Moreover,

(i) if $0\in h\left({\Delta }\right)$, then $h$ is called spirallike (resp., starlike) with respect to an interior point;

(ii) if 0 it is not in $h\left({\Delta }\right)$, then $h$ is called spirallike (resp., starlike) with respect to a boundary point.

In this case, there is a boundary point (say, $z=1$) such that $h\left(1\right):=\angle {lim}_{z\to 1}h\left(z\right)=0$; by symbol $\angle lim$ it is denoted the angular (non-tangential) limit of a function at a boundary point of ${\Delta }$.

The class of spirallike (starlike) functions with respect to a boundary point normalized by the conditions $h\left(1\right)=0$ and $h\left(0\right)=1$ will be denoted by Spiral[1] (resp., Star[1]).

It is known that for any function $h\in \text{Spiral}\left[1\right]$, the limit

$\angle \underset{z\to 1}{lim}\frac{\left(z-1\right){h}^{\text{'}}\left(z\right)}{h\left(z\right)}=\mu$

exists with $\mu \in {\Omega }$, where ${\Omega }=\left\{\lambda :\lambda -1|\le 1$, $\lambda \ne 0\right\}·$

For given $\mu \in {\Omega }$, the class of spirallike functions which satisfy the above relation, will be denoted by ${\text{Spiral}}_{\mu }\left[1\right]$. If the number $\mu$ is real, that is, $0<\mu \le 2$, then the function $h$ is, in fact, starlike. In this case, we can write $h\in {\text{Star}}_{\mu }\left[1\right]$.

We can say that a univalent function $h$ belongs to the class ${\text{Spiral}}_{\mu ,\nu }\left[1,-1\right]$ if it is of the class ${\text{Spiral}}_{\mu }\left[1\right]$ and satisfies the conditions $\angle {lim}_{z\to -1}h\left(z\right)=\infty$, $\angle {lim}_{z\to -1}\frac{{h}^{\text{'}}\left(z\right)\left(z+1\right)}{h\left(z\right)}=\nu \ne 0$. It is know that if $\mu$ is a real number (i.e. $0<\mu \le 2$), then $\nu$ is so and $-\mu \le \nu <0$. Since in this case $h$ is starlike, we can write $h\in {\text{Star}}_{\mu ,\nu }\left[1,-1\right]$.

The authors study the angle bounds for starlike and spirallike functions with respect to a boundary point.

We say that a univalent function $h\in \text{Hol}\left({\Delta },𝒞\right)$ normalized by $h\left(0\right)=0$ is convex in the positive direction of the real axis if for each $z\in {\Delta }$ and $t>0$, $h\left(z\right)+t\in h\left({\Delta }\right)$, ${lim}_{t\to \infty }{h}^{-1}\left(h\left(z\right)+t\right)=1$. The authors find the maximal width size of the image for functions of this class using an angular limit characteristic of functions under consideration. In other words, given such a function, they find the minimal horizontal strip which contains its image. The following question is also natural but more complicated: characterize those functions convex in the positive direction of the real axis whose images contain a whole (two-sided) strip and find the size (width) of this strip. The authors solve this problem for functions having maximal horizontal strips of finite size. The problem is still open for the general case.

##### MSC:
 30C45 Special classes of univalent and multivalent functions
##### Keywords:
angle bounds; functions convex in one direction