Strongly Starlike Functions and Related Classes

Journal of Mathematical Analysis and Applications, 2003

The object of this paper is to prove the sufficiency of a recently established necessary condition for a univalent function to be starlike with respect to a boundary point.

Let A be the class of functions f (z) which are analytic in the open unit disk D and normalized by f (0) = f (0) − 1 = 0. In this paper the inequality |f (z) − 1| ≤ λ • |b(z)|, z ∈ D, is studied for the case b(0) = 0, and sharp sufficient conditions for strong starlikeness of some order are given. This work is an extension of a previous work by Rønning, Ruscheweyh and Samaris where the case b(0) = 0 was covered.

arXiv: Complex Variables, 2018

By using the method of differential subordination, we study a certain subclass of strongly starlike functions which is denoted by $\mathcal{S}^*_t(\alpha_1,\alpha_2)$, including of all normalized and analytic functions satisfying the following two--sided inequality: \begin{equation*} -\frac{\pi\alpha_1}{2}< \arg\left\{\frac{zf'(z)}{f(z)}\right\} <\frac{\pi\alpha_2}{2} \quad |z|<1, \end{equation*} where $0<\alpha_1,\alpha_2\leq1$. The object of the present paper is to derive some certain inequalities for the desired class $\mathcal{S}^*_t(\alpha_1,\alpha_2)$.

Let G denote the class of locally univalent normalized analytic functions f in the unit disk |z| < 1 satisfying the condition Re 1 + zf (z) f (z) < 3 2 for |z| < 1. In this paper, we show in particular that each partial sum s n (z) of f ∈ G is starlike in the disk |z| ≤ 1/2 for n ≥ 12. We also prove that if f ∈ G then Re (s n (z)) > 0 holds in |z| ≤ 1/2 for n ≥ 13.

2001

arXiv: Complex Variables, 2018

Let $\mathcal{S}^*_t(\alpha_1,\alpha_2)$ denote the class of functions $f$ analytic in the open unit disc $\Delta$, normalized by the condition $f(0)=0=f&#39;(0)-1$ and satisfying the following two--sided inequality: \begin{equation*} -\frac{\pi\alpha_1}{2}&lt; \arg\left\{\frac{zf&#39;(z)}{f(z)}\right\} &lt;\frac{\pi\alpha_2}{2} \quad (z\in\Delta), \end{equation*} where $0&lt;\alpha_1,\alpha_2\leq1$. The class $\mathcal{S}^*_t(\alpha_1,\alpha_2)$ is a subclass of strongly starlike functions of order $\beta$ where $\beta=\max\{\alpha_1,\alpha_2\}$. The object of the present paper is to derive some certain inequalities including (for example), upper and lower bounds for ${\rm Re}\{zf&#39;(z)/f(z)\}$, growth theorem, logarithmic coefficient estimates and coefficient estimates for functions $f$ belonging to the class $\mathcal{S}^*_t(\alpha_1,\alpha_2)$.

Applied Mathematics Letters, 2011

Radius of starlikeness of order α Radius of convexity of order α a b s t r a c t Let S * (q c ), c ∈ (0, 1], denote the class of analytic functions f in the unit disc U normalized by f (0) = f ′ (0) − 1 = 0 and satisfying the condition

Asian-European Journal of Mathematics, 2021

A starlike univalent function [Formula: see text] is characterized by [Formula: see text]; several subclasses of starlike functions were studied in the past by restricting [Formula: see text] to take values in a region [Formula: see text] on the right-half plane, or, equivalently, by requiring [Formula: see text] to be subordinate to the corresponding mapping of the unit disk [Formula: see text] to the region [Formula: see text]. The mappings [Formula: see text], [Formula: see text], defined by [Formula: see text] and [Formula: see text] map the unit disk [Formula: see text] to certain nice regions in the right-half plane. For normalized analytic functions [Formula: see text] with [Formula: see text] and [Formula: see text] are subordinate to the function [Formula: see text] for some analytic functions [Formula: see text] and [Formula: see text], we determine the sharp radius for them to belong to various subclasses of starlike functions.

Hokkaido Mathematical Journal, 1998

In this paper we consider starlikeness of the class of functions f(z)=z+ a_{2}z^{2}+\cdot\cdot. which are analytic in the unit disc and satisfy the condition |f'(z)(\frac{z}{f(z)})^{1+\mu}-1|<\lambda , 0<\mu<1,0<\lambda<1 .

We obtain some tests for the starlikeness of analytic functions in terms of their Taylor coefficients. This leads to an improvement of several relevant results about hypergeometric functions. The approach is extended for other well-known subclasses of univalent functions.