scholarly journals Starlikeness associated with parabolic regions

2005 ◽  
Vol 2005 (4) ◽  
pp. 561-570 ◽  
Author(s):  
Rosihan M. Ali
Keyword(s):  
Unit Disk ◽  
Sharp Estimate ◽  
Starlike Functions ◽  
Starlike Function ◽  
Extremal Functions ◽  
Sharp Growth ◽  
Coefficient Problems ◽  
Distortion Theorems ◽  
Convolution Properties ◽  
The Right

A parabolic starlike functionfof orderρin the unit disk is characterized by the fact that the quantityzf′(z)/f(z)lies in a given parabolic region in the right half-plane. Denote the class of such functions byPS∗(ρ). This class is contained in the larger class of starlike functions of orderρ. Subordination results forPS∗(ρ)are established, which yield sharp growth, covering, and distortion theorems. Sharp bounds for the first four coefficients are also obtained. There exist different extremal functions for these coefficient problems. Additionally, we obtain a sharp estimate for the Fekete-Szegö coefficient functional and investigate convolution properties forPS∗(ρ).

scholarly journals Some General Classes of q-Starlike Functions Associated with the Janowski Functions

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 292 ◽  
Author(s):  
Hari Srivastava ◽  
Muhammad Tahir ◽  
Bilal Khan ◽  
Qazi Ahmad ◽  
Nazar Khan
Keyword(s):  
Unit Disk ◽  
Starlike Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Janowski Functions ◽  
Distortion Theorems ◽  
The Relationship

By making use of the concept of basic (or q-) calculus, various families of q-extensions of starlike functions of order α in the open unit disk U were introduced and studied from many different viewpoints and perspectives. In this paper, we first investigate the relationship between various known classes of q-starlike functions that are associated with the Janowski functions. We then introduce and study a new subclass of q-starlike functions that involves the Janowski functions. We also derive several properties of such families of q-starlike functions with negative coefficients including (for example) distortion theorems.

Coefficient inequalities for univalent starlike functions

2013 ◽  
Vol 63 (5) ◽  
Author(s):  
Milutin Obradović ◽  
Saminathan Ponnusamy
Keyword(s):  
Unit Disk ◽  
Complex Number ◽  
Convex Functions ◽  
Analytic Functions ◽  
Starlike Functions ◽  
Right Hand ◽  
Coefficient Inequalities ◽  
The Right ◽  
Necessary And Sufficient

AbstractLet A be the class of analytic functions in the unit disk $$\mathbb{D}$$ with the normalization f(0) = f′(0) − 1 = 0. In this paper the authors discuss necessary and sufficient coefficient conditions for f ∈ A of the form $$\left( {\frac{z} {{f(z)}}} \right)^\mu = 1 + b_1 z + b_2 z^2 + \ldots$$ to be starlike in $$\mathbb{D}$$ and more generally, starlike of some order β, 0 ≤ β < 1. Here µ is a suitable complex number so that the right hand side expression is analytic in $$\mathbb{D}$$ and the power is chosen to be the principal power. A similar problem for the class of convex functions of order β is open.

scholarly journals Radius of starlikeness for some classes containing non-univalent functions

2021 ◽  
pp. 2250009
Author(s):  
Shalu Yadav ◽  
Kanika Sharma ◽  
V. Ravichandran
Keyword(s):  
Unit Disk ◽  
Half Plane ◽  
Univalent Function ◽  
Analytic Functions ◽  
Univalent Functions ◽  
Starlike Functions ◽  
Radius Of Starlikeness ◽  
The Past ◽  
The Right ◽  
Starlike Univalent Function

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.

Coefficient functionals and radius problems of certain starlike functions

2021 ◽  
pp. 2250089
Author(s):  
Adiba Naz ◽  
Sushil Kumar ◽  
V. Ravichandran
Keyword(s):  
Unit Disk ◽  
Half Plane ◽  
Analytic Functions ◽  
Starlike Functions ◽  
Hankel Determinant ◽  
Radius Problems ◽  
Second Hankel Determinant ◽  
The Right ◽  
Inverse Functions

Ma–Minda class (of starlike functions) consists of normalized analytic functions [Formula: see text] defined on the unit disk for which the image of the function [Formula: see text] is contained in some starlike region lying in the right-half plane. In this paper, we obtain the best possible bounds on some initial coefficients for the inverse functions of Ma–Minda starlike functions. Further, the bounds on the Fekete–Szegö functional and the second Hankel determinant are computed for such functions. In addition, some sharp radius estimates are also determined.

ON SOME PROPERTIES OF A GENERALIZED CLASS OF CLOSE-TO-STARLIKE FUNCTIONS

2019 ◽  
Vol 4 (1) ◽  
pp. 193
Author(s):  
Ajab Bai Akbarally ◽  
Nor Siti Khadijah
Keyword(s):  
Univalent Functions ◽  
Starlike Functions ◽  
Starlike Function ◽  
Coefficient Bounds ◽  
New Class ◽  
Distortion Theorems ◽  
Growth And Distortion Theorems

In this paper, we consider a new class of close-to-starlike functions  defined by the Carlson-Shaffer operator. Let denote the class of analytic univalent functions defined by then  ifsatisfy the condition  ,where  and is a starlike function. Properties  of the class  such as the coefficient bounds, growth and distortion theorems and radius properties are investigated. 

scholarly journals Piggyback-Dualitäten

1985 ◽  
Vol 32 (1) ◽  
pp. 1-32 ◽  
Author(s):  
B.A. Davey ◽  
H. Werner
Keyword(s):  
Series Expansion ◽  
Analytic Functions ◽  
Laurent Series ◽  
Integral Operators ◽  
Starlike Functions ◽  
Coefficient Estimates ◽  
Radius Of Convexity ◽  
Distortion Theorems ◽  
Convolution Properties

For the class of meromorphically starlike functions of prescribed order, the concept of type has been introduced. A characterization of meromorphically starlike functions of order α and type β has been obtained when the coefficients in its Laurent series expansion about the origin are all positive. This leads to a study of coefficient estimates, distortion theorems, radius of convexity estimates, integral operators, convolution properties et cetera for this class. It is seen that the class considered demonstrates, in some respects, properties analogous to those possessed by the corresponding class of univalent analytic functions with negative coefficients.

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].

scholarly journals A note on successive coefficients of spirallike functions

Filomat ◽  
2018 ◽  
Vol 32 (4) ◽  
pp. 1199-1207 ◽  
Author(s):  
Ming Li
Keyword(s):  
Unit Disk ◽  
Univalent Functions ◽  
Starlike Functions ◽  
Upper Bounds ◽  
Extremal Functions ◽  
Lower And Upper Bounds ◽  
Bieberbach Conjecture ◽  
Precise Form ◽  
Spirallike Functions ◽  
Class Of Functions

Even there were several facts to show that ||an+1(f)|-|an(f)|| ? 1 is not true for the whole class of normalised univalent functions in the unit disk with with the form f(z) = z + ??,k=2 akzk. In 1978, Leung[7] proved ||an+1(f)|-|an(f)|| is actually bounded by 1 for starlike functions and by this result it is easy to get the conclusion |an| ? n for starlike functions. Since ||an+1(f)|-|an(f)|| ? 1 implies the Bieberbach conjecture (now the de Brange theorem), so it is still interesting to investigate the bound of ||an+1(f)|-|an(f)|| for the class of spirallike functions as this class of functions is closely related to starlike functions. In this article we prove that this functional is bounded by 1 and equality occurs only for the starlike case. We are also able to give a precise form of extremal functions. Furthermore we also try to find the sharp bound of ||an+1(f)|-|an(f)|| for non-starlike spirallike functions. By using the Carath?odory-Toeplitz theorem, we obtain the sharp lower and upper bounds of |an+1(f)|-|an(f)| for n = 1 and n = 2. These results disprove the expected inequality ||an+1(f)|-|an(f)||? cos ? for ?-spirallike functions.

scholarly journals Meromorphic univalent functions with positive coefficients

1985 ◽  
Vol 32 (2) ◽  
pp. 161-176 ◽  
Author(s):  
M.L. Mogra ◽  
T.R. Reddy ◽  
O.P. Juneja
Keyword(s):  
Univalent Functions ◽  
Integral Operators ◽  
Starlike Functions ◽  
Coefficient Estimates ◽  
Radius Of Convexity ◽  
Distortion Theorems ◽  
Convolution Properties ◽  
Positive Coefficients ◽  
Meromorphic Univalent Functions

For the class of meromorphically starlike functions of prescribed order, the concept of type has been introduced. A characterization of meromorphically starlike functions of order α and type β has been obtained when the coefficients in its Laurent series expansion about the origin are all positive. This leads to a study of coefficient estimates, distortion theorems, radius of convexity estimates, integral operators, convolution properties et cetera for this class. It is seen that the class considered demonstrates, in some respects, properties analogous to those possessed by the corresponding class of univalent analytic functions with negative coefficients.

scholarly journals On the Fourth Coefficient of the Inverse of a Starlike Function of Positive Order

2020 ◽  
Author(s):  
Toshiyuki Sugawa ◽  
Li-Mei Wang
Keyword(s):  
Extremal Problem ◽  
Unit Disk ◽  
Complex Plane ◽  
Inverse Function ◽  
Starlike Function ◽  
Extremal Functions ◽  
Positive Order ◽  
Sharp Estimates ◽  
The Third ◽  
Straightforward Proof

We consider the inverse function $z=g(w)$ of a (normalized) starlike function $w=f(z)$ of order $\alpha$ on the unit disk of the complex plane with $0&lt;\alpha&lt;1.$ Krzy{\. z}, Libera and Z\l otkiewicz obtained sharp estimates of the second and the third coefficients of $g(w)$ in their 1979 paper. Prokhorov and Szynal gave sharp estimates of the fourth coefficient of $g(w)$ as a consequence of the solution to an extremal problem in 1981. We give a straightforward proof of the estimate of the fourth coefficient of $g(w)$ together with explicit forms of the extremal functions.

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