scholarly journals Radius of starlikeness of certain analytic functions

2021 ◽  
Vol 71 (1) ◽  
pp. 83-104
Author(s):  
Asha Sebastian ◽  
V. Ravichandran
Keyword(s):  
Analytic Function ◽  
Analytic Functions ◽  
Starlike Functions ◽  
Positive Real Part ◽  
Open Unit ◽  
Open Unit Disk ◽  
Sine Function ◽  
Positive Real ◽  
Radius Of Starlikeness ◽  
Lemniscate Of Bernoulli

Abstract This paper studies analytic functions f defined on the open unit disk of the complex plane for which f/g and (1 + z)g/z are both functions with positive real part for some analytic function g. We determine radius constants of these functions to belong to classes of strong starlike functions, starlike functions of order α, parabolic starlike functions, as well as to the classes of starlike functions associated with lemniscate of Bernoulli, cardioid, lune, reverse lemniscate, sine function, exponential function and a particular rational function. The results obtained are sharp.

scholarly journals Starlikeness of certain non-univalent functions

2021 ◽  
Vol 11 (4) ◽  
Author(s):  
Adam Lecko ◽  
V. Ravichandran ◽  
Asha Sebastian
Keyword(s):  
Analytic Functions ◽  
Univalent Functions ◽  
Starlike Functions ◽  
Positive Real Part ◽  
The Other ◽  
Open Unit ◽  
Sigmoid Function ◽  
Open Unit Disk ◽  
Positive Real ◽  
Lemniscate Of Bernoulli

AbstractWe consider three classes of functions defined using the class $${\mathcal {P}}$$ P of all analytic functions $$p(z)=1+cz+\cdots $$ p ( z ) = 1 + c z + ⋯ on the open unit disk having positive real part and study several radius problems for these classes. The first class consists of all normalized analytic functions f with $$f/g\in {\mathcal {P}}$$ f / g ∈ P and $$g/(zp)\in {\mathcal {P}}$$ g / ( z p ) ∈ P for some normalized analytic function g and $$p\in {\mathcal {P}}$$ p ∈ P . The second class is defined by replacing the condition $$f/g\in {\mathcal {P}}$$ f / g ∈ P by $$|(f/g)-1|<1$$ | ( f / g ) - 1 | < 1 while the other class consists of normalized analytic functions f with $$f/(zp)\in {\mathcal {P}}$$ f / ( z p ) ∈ P for some $$p\in {\mathcal {P}}$$ p ∈ P . We have determined radii so that the functions in these classes to belong to various subclasses of starlike functions. These subclasses includes the classes of starlike functions of order $$\alpha $$ α , parabolic starlike functions, as well as the classes of starlike functions associated with lemniscate of Bernoulli, reverse lemniscate, sine function, a rational function, cardioid, lune, nephroid and modified sigmoid function.

scholarly journals Radius of Star-Likeness for Certain Subclasses of Analytic Functions

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2448
Author(s):  
Caihuan Zhang ◽  
Mirajul Haq ◽  
Nazar Khan ◽  
Muhammad Arif ◽  
Khurshid Ahmad ◽  
...  
Keyword(s):  
Analytic Function ◽  
Unit Disk ◽  
Analytic Functions ◽  
Inverse Function ◽  
Open Unit ◽  
Open Unit Disk ◽  
Sine Function ◽  
Exponential Functions ◽  
Lemniscate Of Bernoulli ◽  
Rotation Function

In this paper, we investigate a normalized analytic (symmetric under rotation) function, f, in an open unit disk that satisfies the condition ℜfzgz>0, for some analytic function, g, with ℜz+1−2nzgz>0,∀n∈N. We calculate the radius constants for different classes of analytic functions, including, for example, for the class of star-like functions connected with the exponential functions, i.e., the lemniscate of Bernoulli, the sine function, cardioid functions, the sine hyperbolic inverse function, the Nephroid function, cosine function and parabolic star-like functions. The results obtained are sharp.

scholarly journals Sufficiency Criteria for q -Starlike Functions Associated with Cardioid

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Saira Zainab ◽  
Ayesha Shakeel ◽  
Muhammad Imran ◽  
Nazeer Muhammad ◽  
Hira Naz ◽  
...  
Keyword(s):  
Analytic Function ◽  
Unit Disk ◽  
Analytic Functions ◽  
Sufficient Conditions ◽  
Starlike Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Image Domain ◽  
Lemniscate Of Bernoulli ◽  
Differential Subordinations

This article deals with the q -differential subordinations for starlike functions associated with the lemniscate of Bernoulli and cardioid domain. The primary goal of this work is to find the conditions on γ for 1 + γ z ∂ q   h z / h n   z   ≺ 1 + z , where h z is analytic function and is subordinated by the function which is producing cardioid domain as its image domain while mapping the open unit disk. Along with this, certain sufficient conditions for q -starlikeness of analytic functions are determined.

scholarly journals Some characteristic properties of analytic functions

2016 ◽  
Vol 24 (1) ◽  
pp. 353-369
Author(s):  
R. K. Raina ◽  
Poonam Sharma ◽  
G. S. Sălăgean
Keyword(s):  
Convex Function ◽  
Unit Disk ◽  
Analytic Functions ◽  
Positive Real Part ◽  
Integral Representations ◽  
Open Unit ◽  
Open Unit Disk ◽  
Positive Real ◽  
Bilinear Mapping ◽  
Initial Coefficient

AbstractIn this paper, we consider a class L(λ, μ; ϕ) of analytic functions f defined in the open unit disk U satisfying the subordination condition that,where is the Sălăgean operator and ϕ(z) is a convex function with positive real part in U. We obtain some characteristic properties giving the coefficient inequality, radius and subordination results, and an inclusion result for the above class when the function ϕ(z) is a bilinear mapping in the open unit disk. For these functions f (z) ; sharp bounds for the initial coefficient and for the Fekete-Szegö functional are determined, and also some integral representations are given.

scholarly journals Generalizing Certain Analytic Functions Correlative to the n-th Coefficient of Certain Class of Bi-Univalent Functions

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Hameed Ur Rehman ◽  
Maslina Darus ◽  
Jamal Salah
Keyword(s):  
Unit Disk ◽  
Analytic Functions ◽  
Univalent Functions ◽  
Positive Real Part ◽  
Polynomial Expansion ◽  
Open Unit ◽  
Open Unit Disk ◽  
Positive Real ◽  
Faber Polynomial ◽  
Coefficient Problems

In the present paper, the authors implement the two analytic functions with its positive real part in the open unit disk. New types of polynomials are introduced, and by using these polynomials with the Faber polynomial expansion, a formula is structured to solve certain coefficient problems. This formula is applied to a certain class of bi-univalent functions and solve the n -th term of its coefficient problems. In the last section of the article, several well-known classes are also extended to its n -th term.

scholarly journals Disk-like functions

1970 ◽  
Vol 11 (2) ◽  
pp. 251-256
Author(s):  
Richard J. Libera
Keyword(s):  
Unit Disk ◽  
Geometric Property ◽  
Starlike Functions ◽  
Positive Real Part ◽  
Open Unit ◽  
Open Unit Disk ◽  
Positive Real ◽  
Image Position

The class s of functions f(z) which are regular and univalent in the open unit disk △ = {z: |z| < 1} each normalized by the conditionshas been studied intensively for over fifty years. A large and very successful portion of this work has dealt with subclasses of L characterized by some geometric property of f[Δ], the image of Δ under f(z), which is expressible in analytic terms. The class of starlike functions in L is one of these [3]; f(z) is starlike with respect to the origin if the segment [0,f(z)] is in f[Δ] for every z in Δ and this condition is equivalent to requiring that have a positive real part in Δ.

scholarly journals On q-ANALOGUE of Differential Subordination Associated with Lemniscate of Bernoulli

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Mohsan Raza ◽  
Hira Naz ◽  
Sarfraz Nawaz Malik ◽  
Sahidul Islam
Keyword(s):  
Analytic Function ◽  
Unit Disk ◽  
Analytic Functions ◽  
Sufficient Conditions ◽  
Differential Subordination ◽  
Open Unit ◽  
Sufficient Condition ◽  
Open Unit Disk ◽  
Lemniscate Of Bernoulli

This article comprises the study of differential subordination with analogue of q -derivative. It includes the sufficient condition on γ for 1 + γ ∂ z q h z / h n z to be subordinated by 1 + A z / 1 + B z , − 1 ≤ B < A ≤ 1 , and implies that h z ≺ 1 + z , where h z is the analytic function in the open unit disk. Moreover, certain sufficient conditions for q -starlikeness of analytic functions related with lemniscate of Bernoulli are determined.

Starlike functions associated with a lune

2017 ◽  
Vol 10 (04) ◽  
pp. 1750064 ◽  
Author(s):  
Shweta Gandhi ◽  
V. Ravichandran
Keyword(s):  
Analytic Function ◽  
Unit Disk ◽  
Half Plane ◽  
Complex Plane ◽  
Starlike Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Lemniscate Of Bernoulli ◽  
The Right

Several subclasses of starlike functions are associated with regions in the right half plane of the complex plane, like half-plane, disks, sectors, parabolas and lemniscate of Bernoulli. For a normalized analytic function [Formula: see text] defined on the open unit disk [Formula: see text] belonging to certain well-known classes of functions associated with the above regions, we investigate the radius [Formula: see text] such that, for the function [Formula: see text], [Formula: see text] lies in the lune defined by [Formula: see text] for all [Formula: see text].

A subclass of starlike functions associated with left-half of the lemniscate of Bernoulli

2014 ◽  
Vol 25 (09) ◽  
pp. 1450090 ◽  
Author(s):  
Rajni Mendiratta ◽  
Sumit Nagpal ◽  
V. Ravichandran
Keyword(s):  
Unit Disk ◽  
Analytic Functions ◽  
Starlike Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Geometric Properties ◽  
Left Half ◽  
Lemniscate Of Bernoulli

Let [Formula: see text] denote the class of all analytic functions f in the open unit disk with the normalizations f(0) = f′(0) - 1 = 0 such that zf′(z)/f(z) lies in the interior of the left-half of the shifted lemniscate of Bernoulli [Formula: see text]. In this paper, we investigate the geometric properties of functions in this class and compute the [Formula: see text]-radius for functions belonging to several interesting classes.

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.

Sign in / Sign up

Export Citation Format

Share Document