scholarly journals Correspondence between spirallike functions and starlike functions

2011 ◽  
Vol 285 (2-3) ◽  
pp. 322-331 ◽  
Author(s):  
Yong Chan Kim ◽  
Toshiyuki Sugawa
Keyword(s):  
Starlike Functions ◽  
Spirallike Functions

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 On a class of functions unifying the classes of Paatero, Robertson and others

1988 ◽  
Vol 11 (2) ◽  
pp. 251-258 ◽  
Author(s):  
S. Bhargava ◽  
S. Nanjunda Rao
Keyword(s):  
Variational Principle ◽  
Schwarzian Derivative ◽  
Starlike Functions ◽  
Class A ◽  
Boundary Rotation ◽  
Alpha Beta ◽  
Spirallike Functions ◽  
Convex And Starlike Functions ◽  
Number Of Classes ◽  
Class Of Functions

We study a classMkλ(α,β,b,c)of analytic functions which unifies a number of classes studied previously by Paatero, Robertson, Pinchuk, Moulis, Mocanu and others. Thus our class includes convex and starlike functions of orderβ, spirallike functions of orderβand functions for whichzf′is spirallike of orderβ, functions of boundary rotation utmostkπ,α-convex functions etc. An integral representation of Paatero and a variational principle of Robertson for the classVkof functions of bounded boundary rotation, yield some representation theorems and a variational principle for our class. A consequence of these basic theorems is a theorem for this classMkλ(α,β,b,c)which unifies some earlier results concerning the radii of convexity of functions in the classVkλ(β)of Moulis and those concerning the radii of starlikeness of functions in the classesUkof Pinchuk andU2(β)of Robertson etc. By applying an estimate of Moulis concerning functions inVkλ(0), we obtain an inequality in the classMkλ(α,β,b,c)which will contain an estimate for the Schwarzian derivative of functions in the classVkλ(β)and in particular the estimate of Moulis for the Schwarzian of functions inVkλ(0).

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 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.

On λ-pseudo bi-starlike functions related to some conic domains

2020 ◽  
Vol 61(12) (2) ◽  
pp. 381-392
Author(s):  
Gangadhara Murugusundaramoorthy ◽  
◽  
Janusz Sokol ◽  
Keyword(s):  
Starlike Functions

A Class of Harmonic Starlike Functions defined by Multiplier Transformation

2020 ◽  
pp. 455-469
Author(s):  
Deepali Khurana ◽  
Raj Kumar ◽  
Sibel Yalcin
Keyword(s):  
Convex Combination ◽  
Univalent Functions ◽  
Extreme Points ◽  
Starlike Functions ◽  
Harmonic Mappings ◽  
Distortion Bounds ◽  
Radius Of Convexity ◽  
Univalent Harmonic Mappings ◽  
Harmonic Starlike ◽  
Multiplier Transformation

We define two new subclasses, $HS(k, \lambda, b, \alpha)$ and \linebreak $\overline{HS}(k, \lambda, b, \alpha)$, of univalent harmonic mappings using multiplier transformation. We obtain a sufficient condition for harmonic univalent functions to be in $HS(k,\lambda,b,\alpha)$ and we prove that this condition is also necessary for the functions in the class $\overline{HS} (k,\lambda,b,\alpha)$. We also obtain extreme points, distortion bounds, convex combination, radius of convexity and Bernandi-Libera-Livingston integral for the functions in the class $\overline{HS}(k,\lambda,b,\alpha)$.

The sharp bounds of the second and third Hankel determinants for the class 𝓢𝓛*

2020 ◽  
Vol 70 (4) ◽  
pp. 849-862
Author(s):  
Shagun Banga ◽  
S. Sivaprasad Kumar
Keyword(s):  
Triangle Inequality ◽  
Starlike Functions ◽  
Sharp Bound ◽  
The Novel ◽  
Sharp Bounds ◽  
Hankel Determinants ◽  
Lemniscate Of Bernoulli ◽  
Carathéodory Functions ◽  
The Right

AbstractIn this paper, we use the novel idea of incorporating the recently derived formula for the fourth coefficient of Carathéodory functions, in place of the routine triangle inequality to achieve the sharp bounds of the Hankel determinants H3(1) and H2(3) for the well known class 𝓢𝓛* of starlike functions associated with the right lemniscate of Bernoulli. Apart from that the sharp bound of the Zalcman functional: $\begin{array}{} |a_3^2-a_5| \end{array}$ for the class 𝓢𝓛* is also estimated. Further, a couple of interesting results of 𝓢𝓛* are also discussed.

scholarly journals On a class of analytic functions related to Robertson’s formula and subordination

2021 ◽  
Vol 27 (1) ◽  
Author(s):  
Adam Lecko ◽  
Gangadharan Murugusundaramoorthy ◽  
Srikandan Sivasubramanian
Keyword(s):  
Integral Representation ◽  
Analytic Functions ◽  
Boundary Point ◽  
Unit Disc ◽  
Starlike Functions ◽  
Analytic Formula ◽  
Growth Theorem

AbstractIn this paper, we define and study a class of analytic functions in the unit disc by modification of the well-known Robertson’s analytic formula for starlike functions with respect to a boundary point combined with subordination. An integral representation and growth theorem are proved. Early coefficients and the Fekete–Szegö functional are also estimated.

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