Coecient Functional and Bohr-Rogosinski Phenomenon for Analytic functions involving Semigroup Generators Surya Giri and S. Sivaprasad Kumar

2025-04-29 0 0 391.01KB 15 页 10玖币

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Coeﬃcient Functional and Bohr-Rogosinski Phenomenon for

Analytic functions involving Semigroup Generators

Surya Giri and S. Sivaprasad Kumar

Abstract

This paper examines the coeﬃcient problems for the class of semigroup generators, a topic in complex

dynamics that has recently been studied in context of geometric function theory. Further, sharp bounds

of coeﬃcient functional such as second order Hankel determinant, third order Toeplitz and Hermitian-

Toeplitz determinants are derived. Additionally, the sharp growth estimates and the bounds of diﬀerence

of successive coeﬃcients are determined, which are used to prove the Bohr and the Bohr- Rogosinski

phenomenon for the class of semigroup generators.

Keywords: Holomorphic generators; Hankel determinant; Toeplitz determinant; Zalcman functional; Succes-

sive coeﬃcient diﬀerence; Bohr and Bohr-Rogosinski radius.

AMS Subject Classiﬁcation: 30C45, 30C50, 30C55, 47H20, 37L05.

1 Introduction

Let Hbe the class of holomorphic functions in the unit disk Dand A⊂ H containing functions of the form

f(z) = z+∞

n=2

anzn.(1.1)

By B, we denote the class of holomorphic self mappings from Dto D. A family {ut(z)}t≥0⊂ B is called a one

parameter continuous semigroup if (i) limt→0ut(z) = z, (ii)ut+s(z) = ut(z)◦us(z), and (iii) limt→sut(z) =

us(z) for each z∈Dhold.

Berkson and Porta [3] showed that each one parameter semigroup is locally diﬀerentiable in parameter

t≥0 and moreover, if

lim

t→0

z−ut(z)

t=f(z),

which is a holomorphic function, then ut(z) is the solution of the the Cauchy problem

∂ut(z)

∂t +f(ut(z)) = 0, u0(z) = z.

The function fis called the inﬁnitesimal (holomorphic) generator of semigroup {ut(z)}⊂B. The class of all

holomorphic generators is denoted by G.Also, note that each element of {ut(z)}generated by f∈ H(D,C)

is univalent function while fis not necessarily univalent [13]. Diﬀerent analytic criteria are available to show

that a function is semigroup generator [3, 37, 6]. Berkson and Porta [3] proved:

Theorem 1.1. The following assertions are equivalent:

(a)f∈ G;

(b)f(z)=(z−σ)(1 −z¯σ)p(z)with some σ∈Dand p∈ H,Re(p(z)) ≥0.

arXiv:2210.13166v1 [math.CV] 24 Oct 2022

The point σ∈D:= {z∈C:|z| ≤ 1}is unique and called the Denjoy–Wolﬀ point of the semigroup

generated by f(z). By Denjoy-Wolﬀ theorem [13, 37, 36] for continuous semigroup, limt→∞ u(t, z) = σif

the semigroup generated by fis neither an elliptic automorphism of Dnor an identity map for at least one

t∈[0,∞). We denote the class of inﬁnitesimal generators with Denjoy-Wolﬀ point σby G[σ].For σ= 0, we

obtain the following subclass

G[0] = {f∈ G :f(z) = zp(z),Re p(z)≥0}.

Bracci et al. [7] considered the class G0=G[0]∩A. In the study of non-autonomous problem such as Loewner

theory, the class G0plays a signiﬁcant role [8, 12]. Various subclasses of G0are considered with parameter,

which is also called ﬁltration. For instance, the class

Aβ=f∈ A : Re βf(z)

z+ (1 −β)f0(z)>0, β ∈[0,1](1.2)

is a subclass of G0.In [7], the authors proved that Aβ1(Aβ2(G0for 0 ≤β1< β2<1 and whenever

f∈ Aβ,

Re f(z)

z≥Z1

1−t1−β

1 + t1−βdt.

Clearly, when β= 0, the class Aβreduces to the class R:= {f∈ A : Re f0(z)>0},which is called the

class of bounded turning functions. It can be easily seen that R⊂G0and each f∈ R satisﬁes the Noshiro-

Warshawski condition [12], thus every member of Rgenerates a semigroup that is univalent. Elin et al. [14]

solved the radii problems for the class Aβ. They found the radii r∈(0,1) for f∈ Aβsuch that f(rz)/r

belong to the class of starlike functions, denoted by S∗, and some other subclasses of starlike functions. This

problem arises from the fact that neither S∗⊂ Aβnor Aβ⊂ S∗. Generalizing this work, Giri and Kumar

[16] obtained rsuch that f(rz)/r belong to the class of Ma-Minda starlike functions.

Coeﬃcient problems, growth estimates and others were still open for the class Aβ. In this paper, we focus

on these problems. We ﬁnd the bound of nth coeﬃcient of f∈ Aβand coeﬃcient functional such as second

Hankel determinant, third order Toeplitz and Hermitian Toeplitz determinant and Zalcman functional.

Later, Bohr and Bohr-Rogosinski phenomenon with growth estimates are also discussed for this class.

In 1914, Bohr [5] proved that, if ω(z) = P∞

n=0 cnzn∈ B, then P∞

n=0|cn|rn≤1 for all z∈Dwith

|z|=r≤1/3.The constant 1/3 is known as Bohr radius. This inequality was ﬁrst derived by Bohr for

r≤1/6 and sharpened independently to r≤1/3 by Wiener, Riesz, and Schur. In recent years, a lot of

study have been carried out on the Bohr inequality for functions, which map Donto other domains, say Ω

[4, 15]. Diﬀerent generalizations of the Bohr inequality are taken into consideration [39, 27]. We say that

Deﬁnition 1.2. The class Aβsatisﬁes the Bohr phenomenon if there exists rbsuch that

|z|+∞

n=2|an||z|n≤d(f(0), ∂f(Ω))

holds in |z|=r≤rb,where ∂f(Ω) is the boundary of image domain of Dunder fand ddenotes the

Euclidean distance between f(0) and ∂f(Ω).

For m= 1, Muhanna [32] showed that the Bohr phenomenon holds for the class of univalent functions

and the class of convex functions, when |z|=r≤3−2√2 and |z|=r≤1/3 respectively. We refer to the

survey article [31] for further details on this topic. There is also the concept of Rogosinski radius along with

the Bohr radius, although a little is known about Rogosinski radius in comparison to Bohr radius [17, 22, 35].

It says that, if ω(z) = P∞

n=0 cnzn∈ B, then

N−1

n=0 |cn||z|n≤1 (N∈N)

in the disk |z|=r≤1/2.The radius 1/2 is called the Rogosinski radius. Kayumov et al. [18] considered

the following expression

N(z) := |f(z)|+∞

n=N|an||z|n

and found the radius rNsuch that Rf

N(z)≤1 in |z|=r≤rNfor the Ces´aro operators on the space of

bounded analytic functions. Here, we say that:

Deﬁnition 1.3. The class Aβsatisﬁes the Bohr-Rogosinski phenomenon if there exist rNsuch that

|f(zm)|+∞

n=N|an||z|n≤d(f(0), ∂f(Ω))

holds in |z|=r≤rN.

Section 5 is devoted to ﬁnd the rband rNfor the class Aβ.

For f(z) = z+P∞

n=2 anzn∈ A, the mth Hankel, Toeplitz and Hermitian Toeplitz determinant for m≥1

and n≥0 are respectively given by

Hm(n)(f) = 

anan+1 ··· an+m−1

an+1 an+2 ··· an+m

an+m−1an+m··· an+2m−2



Tm(n)(f) = 

anan+1 ··· an+m−1

an+1 an··· an+m−2

an+m−1an+m−2··· an



,(1.3)

Tm,n(f) = 

anan+1 ··· an+m−1

¯an+1 an··· an+m−2

¯an+m−1¯an+m−2··· an



,(1.4)

where ¯an=an. Toeplitz matrices have constant entries along their diagonals, while Hankel matrices have

constant entries along their reverse diagonals. In particular,

H2(n)(f) = anan+2 −a2

n+1, T3(1)(f)=1−2a2

2+ 2a2

2a3−a2

and T3,1(f) = 1 −2|a2|2+ 2 Re(a2

2¯a3)− |a3|2.Finding the sharp bound of |H2(2)(f)|for the class Sand its

subclasses has always been the focus of many researchers. Although, investigations concerning Toeplitz and

Hermitian Toeplitz are recently introduced in [2, 11], a summary of some of the more signiﬁcant results is

given in [38]. For more work in this direction (see [21, 20, 24, 33]).

In 1999, Ma [28] proposed a conjecture for f(z) = z+P∞

n=2 anzn∈ S that

|Jm,n|:= |anam−an+m−1| ≤ (m−1)(n−1).

They proved this conjecture for the class of starlike functions and univalent functions with real coeﬃcients. It

is also called generalized Zalcman conjecture as it generalizes the Zalcman conjecture |a2

n−a2n−1| ≤ (2n−1)2

for f∈ S. Recently, bound of |J2,3|are obtained for various subclasses of A[1, 10]. In section 2 and 3, we

obtain the sharp bound of |H2(2)(f)|,|T3(1)(f)|and |J2,3(f)|for f∈ Aβ.

2 Hankel Determinant and Zalcman Functional

Theorem 2.1. If f∈ Aβis of the form (1.1), then

|an| ≤ 2

n−β(n−1).(2.1)

Further, this inequality is sharp for each n.

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CoecientFunctionalandBohr-RogosinskiPhenomenonforAnalyticfunctionsinvolvingSemigroupGeneratorsSuryaGiriandS.SivaprasadKumarAbstractThispaperexaminesthecoecientproblemsfortheclassofsemigroupgenerators,atopicincomplexdynamicsthathasrecentlybeenstudiedincontextofgeometricfunctiontheory.Further,sharpb...

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Coecient Functional and Bohr-Rogosinski Phenomenon for Analytic functions involving Semigroup Generators Surya Giri and S. Sivaprasad Kumar

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