scholarly journals Criteria for Strongly Starlike Functions

ISRN Geometry ◽  
2011 ◽  
Vol 2011 ◽  
pp. 1-8
Author(s):  
Neng Xu
Keyword(s):  
Unit Disk ◽  
Starlike Functions ◽  
Differential Subordination ◽  
Strongly Starlike Functions ◽  
Strongly Starlike ◽  
Differential Subordinations

Let f(z) be analytic in the unit disk U={z:|z|<1} with f(0)=f'(0)-1=0 and (f(z)/z)f'(z)≠0. By using the method of differential subordinations, we determine the largest number α(β,λ,μ,m) such that, for some β,λ,μ, and m, the differential subordination λzf'(z)/f(z)1-μ1+(zf''(z)/f'(z))-zf'(z)/f(z)+zf'(z)/f(z)m≺1+z/1-zα(β,λ,μ,m)(z∈U) implies zf'(z)/f(z)≺1+z/1-zβ. Some useful consequences of this result are also given.

scholarly journals On the Fekete-Szegö problem

2000 ◽  
Vol 24 (9) ◽  
pp. 577-581 ◽  
Author(s):  
B. A. Frasin ◽  
Maslina Darus
Keyword(s):  
Analytic Function ◽  
Unit Disk ◽  
Upper Bound ◽  
Starlike Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Strongly Starlike Functions ◽  
Strongly Starlike

Letf(z)=z+a2z2+a3z3+⋯be an analytic function in the open unit disk. A sharp upper bound is obtained for|a3−μa22|by using the classes of strongly starlike functions of orderβand typeαwhenμ≥1.

scholarly journals Growth Theorems for a Subclass of Strongly Spirallike Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Yan-Yan Cui ◽  
Chao-Jun Wang ◽  
Si-Feng Zhu
Keyword(s):  
Unit Disk ◽  
Complex Plane ◽  
Starlike Functions ◽  
Geometric Properties ◽  
Strongly Starlike Functions ◽  
Strongly Starlike ◽  
Spirallike Functions ◽  
Analytic Mappings

In this paper we consider a subclass of strongly spirallike functions on the unit diskDin the complex planeC, namely, strongly almost spirallike functions of typeβand orderα. We obtain the growth results for strongly almost spirallike functions of typeβand orderαon the unit diskDinCby using subordination principles and the geometric properties of analytic mappings. Furthermore we get the growth theorems for strongly almost starlike functions of orderαand strongly starlike functions on the unit diskDofC. These growth results follow the deviation results of these functions.

scholarly journals Extension of Nunokawa Lemma for Functions with Fixed Second Coefficient and Its Applications

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Mostafa Amani ◽  
Rasoul Aghalary ◽  
Ali Ebadian
Keyword(s):  
Analytic Functions ◽  
Univalent Functions ◽  
Sufficient Conditions ◽  
Starlike Functions ◽  
Differential Subordination ◽  
Differential Inequalities ◽  
Initial Coefficient ◽  
Strongly Starlike Functions ◽  
Strongly Starlike

In this paper, we study some properties of analytic functions with fixed initial coefficients. The methodology of differential subordination is used for modification and improvements of several well-known results for subclasses of univalent functions by restricting the functions with fixed initial coefficients. Actually, by extending the Nunokawa lemma for fixed initial coefficient functions, we obtain some novel results on subclasses of univalent functions, such as differential inequalities for univalency or starlikeness of analytic functions. Also, we provide some new sufficient conditions for strongly starlike functions. The results of this paper extend and improve the previously known results by considering functions with fixed second coefficients.

scholarly journals Differential inequalities for spirallike and strongly starlike functions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Nak Eun Cho ◽  
Oh Sang Kwon ◽  
Young Jae Sim
Keyword(s):  
Analytic Function ◽  
Sufficient Conditions ◽  
Starlike Functions ◽  
Differential Subordination ◽  
Differential Inequalities ◽  
First Order ◽  
Strongly Starlike Functions ◽  
Strongly Starlike ◽  
Spirallike Functions

AbstractIn this paper, by using a technique of the first-order differential subordination, we find several sufficient conditions for an analytic function p such that $p(0)=1$ p ( 0 ) = 1 to satisfy $\operatorname{Re}\{ {\mathrm{e}}^{{\mathrm{i}}\beta } p(z) \} > \gamma $ Re { e i β p ( z ) } > γ or $| \arg \{p(z)-\gamma \} |<\delta $ | arg { p ( z ) − γ } | < δ for all $z\in \mathbb{D}$ z ∈ D , where $\beta \in (-\pi /2,\pi /2)$ β ∈ ( − π / 2 , π / 2 ) , $\gamma \in [0,\cos \beta )$ γ ∈ [ 0 , cos β ) , $\delta \in (0,1]$ δ ∈ ( 0 , 1 ] and $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1 \}$ D : = { z ∈ C : | z | < 1 } . The results obtained here will be applied to find some conditions for spirallike functions and strongly starlike functions in $\mathbb{D}$ D .

scholarly journals Argument es-timates of strongly starlike functions associated with Noor-integral operator

2009 ◽  
Vol 40 (1) ◽  
pp. 7-13
Author(s):  
T. N. Shanmugam ◽  
S. Sivasubramanian ◽  
B. A. Frasin
Keyword(s):  
Integral Operator ◽  
Starlike Functions ◽  
Strongly Starlike Functions ◽  
Strongly Starlike

The purpose of this present paper is to derive some properties of a certain new sub-classes of strongly starlike functions defined by the Noor integral operator. Relevant connections of the results, which are presented in the paper, with various known results are also considered.

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