# Classes of Meromorphic Functions Defined by the Hadamard Product

Authors: J. Dziok
Journal: International Journal of Mathematics and Mathematical Sciences
 (2010)

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## Abstract

The object of the present paper is to introduce new classes of meromorphic functions with
varying argument of coefficients defined by means of the Hadamard product (or convolution). Several properties like the coefficients bounds, growth and distortion theorems, radii of starlikeness and convexity, and partial sums are investigated. Some consequences of the main results for well-known classes of meromorphic functions are also pointed out.


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## Body

## 1. Introduction

Letℳ̃ denote the class of functions which are analytic in 𝒟=𝒟(1), where(1.1)𝒟(r)={z∈ℂ:0<|z|<r},
with a simple pole in the point z=0. By ℳ, we denote the class of functions f∈ℳ̃ of the form(1.2)f(z)=1z+∑n=1∞anzn(z∈𝒟).
Also, by 𝒯ηε(η∈ℝ,ε∈{0,1}), we denote the class of functions f∈ℳ of the form (1.2) for which(1.3)arg(an)=επ-(n+1)η(n∈ℕ:={1,2,3,…}).
For η=0, we obtain the classes 𝒯00 and 𝒯01 of functions with positive coefficients and negative coefficients, respectively.Motivated by Silverman [1], we define the class(1.4)𝒯ε:=⋃η∈ℝ𝒯ηε.It is called the class of functions with varying argument of coefficients.Letα∈〈0,1),r∈(0,1〉. A function f∈ℳ is said to be meromorphically convex of orderα in 𝒟(r) if(1.5)Re(1+zf′′(z)f′(z))<-α(z∈𝒟(r)).A functionf∈ℳ is said to be meromorphically starlike of orderα in 𝒟(r) if(1.6)Re(zf′(z)f(z))<-α(z∈𝒟(r)).
We denote by ℳ𝒮c(α) the class of all functions f∈ℳ, which are meromorphically convex of order α in 𝒟 and by ℳ𝒮*(α), we denote the class of all functions f∈ℳ, which are meromorphically starlike of order α in 𝒟. We also set(1.7)ℳ𝒮c=ℳ𝒮c(0),ℳ𝒮*=ℳ𝒮*(0).
It is easy to show that for a function f∈𝒯η0, the condition (1.6) is equivalent to the following:(1.8)|zf′(z)f(z)+1|<1-α(z∈𝒟(r)).Letℬ be a subclass of the class ℳ. We define the radius of starlikeness of orderα and the radius of convexity of order α for the class ℬ by(1.9)Rα*(ℬ)=inff∈ℬ(sup{r∈(0,1]:fismeromorphicallystarlikeoforderαin𝒟(r)});Rαc(ℬ)=inff∈ℬ(sup{r∈(0,1]:fismeromorphicallyconvexoforderαin𝒟(r)}),
respectively.Let functionsf,g be analytic in 𝒰:=𝒟∪{0}. We say that the function f is subordinate to the function g, and write f(z)≺g(z) (or simply f≺g) if there exists a function ω analytic in 𝒰, |ω(z)|≤|z|(z∈𝒰), such that(1.10)f(z)=g(ω(z))(z∈𝒰).
In particular, if F is univalent in 𝒰, we have the following equivalence:(1.11)f(z)≺F(z)⇔f(0)=F(0),f(𝒰)⊂F(𝒰).For functionsf,g∈ℳ̃ of the form(1.12)f(z)=∑n=-1∞anzn,g(z)=∑n=-1∞bnzn,
by f*g we denote the Hadamard product(orconvolution) of f and g, defined by(1.13)(f*g)(z)=∑n=-1∞anbnzn(z∈𝒟).LetA,B be real parameters, -1≤A<B≤1, and let φ,ϕ be given functions from the class ℳ.By𝒲(ϕ,φ;A,B), we denote the class of functions f∈ℳ such that (φ*f)(z)≠0(z∈𝒟) and(1.14)(ϕ*f)(z)(φ*f)(z)≺1+Az1+Bz.
Moreover, let us define(1.15)𝒯𝒲ε(ϕ,φ;A,B):=𝒯ε∩𝒲(ϕ,φ;A,B),𝒯𝒲ηε(ϕ,φ;A,B):=𝒯ηε∩𝒲(ϕ,φ;A,B),
where the functions φ,ϕ have the form(1.16)ϕ(z)=1z+(-1)ε+1∑n=1∞βnzn,φ(z)=1z+(-1)ε∑n=1∞αnzn(z∈𝒟),
and the sequences {αn},{βn} are nonnegative real, with(1.17)αn+βn>0(n∈ℕ).
Moreover, let us put(1.18)dn:=(1+B)βn+(1+A)αn(n∈ℕ).It is easy to show that(1.19)f(z)∈𝒯𝒲(zϕ′(z),zφ′(z);A,B)⇔-zf′(z)∈𝒯𝒲(ϕ,φ;A,B),f(z)∈𝒯𝒲η0(zϕ′(z),zφ′(z);A,B)⇔-zf′(z)∈𝒯𝒲η1(ϕ,φ;A,B).The object of the present paper is to investigate the coefficient estimates, distortion properties and the radii of starlikeness and convexity, and partial sums for the classes of meromorphic functions with varying argument of coefficients. Some remarks depicting consequences of the main results are also mentioned.


## 2. Coefficients Estimates

First we mention a sufficient condition for functions to belong to the class𝒲(ϕ,φ;A,B).Theorem 2.1.
Let{dn} be defined by (1.18), -1≤A<B≤1. If a function f of the form (1.2) satisfies the condition
(2.1)∑n=1∞dn|an|≤B-A,
then f belongs to the class  𝒲(ϕ,φ;A,B).Proof.
A functionf of the form (1.2) belongs to the class 𝒲ε(ϕ,φ;A,B) if and only if there exists a function ω,|ω(z)|≤|z|(z∈𝒟), such that
(2.2)(ϕ*f)(z)(φ*f)(z)=1+Aω(z)1+Bω(z)(z∈𝒟),
or equivalently
(2.3)|z(ϕ*f)(z)-z(φ*f)(z)Bz(ϕ*f)(z)-Az(φ*f)(z)|<1(z∈𝒟).
Thus, it is sufficient to prove that
(2.4)|z(ϕ*f)(z)-z(φ*f)(z)|-|Bz(ϕ*f)(z)-Az(φ*f)(z)|<0(z∈𝒟).
Indeed, letting |z|=r(0<r<1), we have
(2.5)|z(φ*f)(z)-z(ϕ*f)(z)|-|Bz(ϕ*f)(z)-Az(φ*f)(z)|=|∑n=1∞(βn+αn)anzn+1|-|(B-A)-∑n=1∞(Bβn+Aαn)anzn+1|≤∑n=1∞(βn+αn)|an|rn+1-(B-A)+∑n=1∞(Bβn+Aαn)|an|rn+1≤∑n=1∞dn|an|rn+1-(B-A)<0,
whence f∈𝒲(ϕ,φ;A,B).Theorem 2.2.
Letf be a function of the form (1.2), with (1.3). Then  f belongs to the class  𝒯𝒲ηε(ϕ,φ;A,B) if and only if the condition  (2.1) holds true.Proof.
In view of Theorem2.1, we need only to show that each function f from the class 𝒯𝒲ηε(ϕ,φ;A,B) satisfies the coefficient inequality (2.1). Let f∈𝒯𝒲ηε(ϕ,φ;A,B). Then by (2.3) and (1.2), we have
(2.6)|∑n=1∞(βn+αn)anzn+1B-A-∑n=1∞(-1)ε(Bβn+Aαn)anzn+1|<1(z∈𝒟).
Therefore, putting z=reiη(0≤r<1), and applying (1.3), we obtain
(2.7)∑n=1∞(βn+αn)|an|rn+1B-A-∑n=1∞(Bβn+Aαn)|an|rn+1<1.
It is clear, that the denominator of the left hand said cannot vanish for r∈〈0,1). Moreover, it is positive for r=0, and in consequence for r∈〈0,1). Thus, by (2.7), we have
(2.8)∑n=1∞[(1+B)βn+(1+A)αn]|an|rn+1<B-A,
which, upon letting r→1-, readily yields the assertion (2.1).From Theorem2.2, we obtain coefficients estimates for the class 𝒯𝒲ηε(ϕ,φ;A,B).Corollary 2.3.
If a functionf of the form  (1.2) belongs to the class  𝒯𝒲ηε(ϕ,φ;A,B),  then
(2.9)|an|≤B-Adn(n∈ℕ),
where {dn} is defined by (1.18). The result is sharp. The functions  fn,η of the form
(2.10)fn,η(z)=1z+B-Aei{(n+1)η-επ}dnzn(z∈𝒟;n∈ℕ)
are the extremal functions.


## 3. Distortion Theorems

From Theorem2.2, we have the following lemma.Lemma 3.1.
Let a functionf of the form (1.2) belong to the class 𝒯𝒲ηε(ϕ,φ;A,B). If the sequence  {dn}defined by (1.18) satisfies the inequality
(3.1)d1≤dn(n∈ℕ),
then
(3.2)∑n=1∞an≤B-Ad1.
Moreover, if
(3.3)nd1≤dn(n∈ℕ),
then
(3.4)∑n=1∞nan≤B-Ad1.Theorem 3.2.
Let a functionf belong to the class  𝒯𝒲ηε(ϕ,φ;A,B). If the sequence  {dn} defined by (1.18) satisfies  (3.1), then
(3.5)1r-B-Ad1r≤|f(z)|≤1r+B-Ad1r(|z|=r<1).
Moreover, if  (3.3) holds, then
(3.6)1r2-B-Ad1≤|f′(z)|≤1r2+B-Ad1(|z|=r<1).
The result is sharp, with the extremal function  f1,η of the form (2.10).Proof.
Let a functionf of the form (1.2) belong to the class 𝒯𝒲ηε(ϕ,φ;A,B),|z|=r<1. Since
(3.7)|f(z)|=|1z+∑n=1∞anzn|≤1r+∑n=1∞|an|rn=1r+r∑n=1∞|an|rn-1≤1r+r∑n=1∞|an|,|f(z)|=|z+∑n=1∞anzn|≥1r-∑n=1∞|an|rn=1r-r∑n=1∞|an|rn-1≥1r-r∑n=1∞|an|,
then by Lemma 3.1 we have (3.5). Analogously we prove (3.6).


## 4. The Radii of Convexity and Starlikeness

Theorem 4.1.
The radius of starlikeness of orderα for the class  𝒯𝒲η0(ϕ,φ;A,B) is given by
(4.1)Rα*(𝒯𝒲η0(ϕ,φ;A,B))=infn∈ℕ((1-α)dn(n-α)(B-A))1/(n+1),
where dn is defined by (1.18).Proof.
The functionf∈𝒯η0 of the form (1.2) is meromorphically starlike of order α in the disk 𝒟(r),0<r≤1, if and only if it satisfies the condition (1.8). Since
(4.2)|zf′(z)f(z)+1|=|∑n=1∞(n+1)anzn+11+∑n=1∞anzn+1|≤∑n=1∞(n+1)|an||z|n+11-∑n=1∞|an||z|n+1,
putting |z|=r, the condition (1.8) is true if
(4.3)∑n=1∞n-α1-α|an|rn+1≤1.
By Theorem 2.2, we have
(4.4)∑n=1∞dnB-A|an|≤1.
Thus, the condition (4.3) is true if
(4.5)n-α1-αrn+1≤dnB-A(n∈ℕ),
that is, if
(4.6)r≤((1-α)dn(n-α)(B-A))1/(n+1)(n∈ℕ).
It follows that each function f∈𝒯𝒲η0(ϕ,φ;A,B) is meromorphically starlike of order α in the disk 𝒟(r), where r=R*(𝒯𝒲η0(ϕ,φ;A,B)) is defined by (4.1). Moreover, the radius of starlikeness of the functions fn,η defined by (2.10) is given by
(4.7)rn*=min{1,((1-α)dn(n-α)(B-A))1/(n+1)}(n∈ℕ).
Thus we have (4.1).Theorem 4.2.
The radius of convexity of orderα for the class  𝒯𝒲η1(ϕ,φ;A,B) is given by
(4.8)Rαc(𝒯𝒲η1(ϕ,φ;A,B))=infn∈ℕ((1-α)dnn(n-α)(B-A))1/(n+1),
where dn is defined by (1.18).Proof.
The proof is analogous to that of Theorem4.1, and we omit the details.


## 5. Partial Sums

Letf∈ℳ be a function of the form (1.2). Motivated by Silverman [2] and Silvia [3] (see also [4]), we define the partial sums fm defined by(5.1)fm(z)=1z+∑n=1manzn(m∈ℕ).
In this section, we consider partial sums of functions from the class 𝒯𝒲ηε(ϕ,φ;A,B) and obtain sharp lower bounds for the real part of ratios of f to fm and f′ to fm′.Theorem 5.1.
Letm∈ℕ and let the sequence  {dn}, defined by (1.18), satisfy the inequalities
(5.2)B-A≤dn≤dn+1(m∈ℕ).
If a function f belongs to the class 𝒯𝒲ηε(ϕ,φ;A,B), then
(5.3)Re{f(z)fm(z)}≥1-B-Adm+1(z∈𝒟),(5.4)Re{fm(z)f(z)}≥dm+1B-A+dm+1(z∈𝒟).
The bounds are sharp, with the extremal function  fm+1,η of the form  (2.10).Proof.
Let a functionf of the form (1.2) belong to the class 𝒯𝒲ηε(ϕ,φ;A,B). Then by (5.2) and Theorem 2.2, we have
(5.5)∑n=1m|an|+dm+1B-A∑n=m+1∞|an|≤∑n=1∞dnB-A|an|≤1.
If we put
(5.6)g(z)=dm+1B-A{f(z)fm(z)-(1-B-Adm+1)}=1+(dm+1/(B-A))∑n=m+1∞anzn+11+∑n=1manzn+1(z∈𝒟),
then it suffices to show that
(5.7)Reg(z)≥0(z∈𝒟),
or
(5.8)|g(z)-1g(z)+1|≤1(z∈𝒟).
Applying (5.5), we find that
(5.9)|g(z)-1g(z)+1|≤(dm+1/(B-A))∑n=m+1∞|an|2-2∑n=1n|an|-(dm+1/(B-A))∑n=m+1∞|an|≤1(z∈𝒟),
which readily yields the assertion (5.3). In order to see that f=fm+1,η gives the result sharp, we observe that for z=reiη we have
(5.10)f(z)fm(z)=1-(B-A)rm+2dm+1→r→1-1-B-Adm+1.
Similarly, if we take
(5.11)h(z)=(B-A+dm+1){fm(z)f(z)-dm+1B-A+dm+1}(z∈𝒟),
and make use of (5.5), we can deduce that
(5.12)|h(z)-1h(z)+1|≤(1+(dm+1/(B-A)))∑n=m+1∞|an|2-2∑n=1m|an|-(1-(dm+1/(B-A)))∑n=m+1∞|an|≤1(z∈𝒟),
which leads us immediately to the assertion (5.4). The bound in (5.4) is sharp for each m∈N, with the extremal function f=fm+1,η, given by (2.10).Theorem 5.2.
Letm∈N and let the sequence  {dn}, defined by (1.18), satisfy the inequalities (5.2).  If a function f belongs to the class 𝒯𝒲ηε(ϕ,φ;A,B), then
(5.13)Re{f′(z)fm(z)}≥1-(B-A)(m+1)dm+1(z∈𝒟),Re{fm′(z)f(z)}≥dm+1(B-A)(m+1)+dm+1(z∈𝒟).
The bounds are sharp, with the extremal function  fm+1,η of the form  (2.10).Proof.
The proof  is analogous to that of Theorem5.1, and we omit the details.Remark 5.3.
We observe that the obtained results are true if we replace the class𝒯𝒲ηε(ϕ,φ;A,B) by 𝒯𝒲ε(ϕ,φ;A,B).


## 6. Concluding Remarks

We conclude this paper by observing that, in view of the subordination relation (1.18), by choosing the functions ϕ and φ, we can define new classes of functions. In particular, the class(6.1)𝒲(φ;A,B):=𝒲(-zφ′(z),φ(z);A,B)
contains functions f∈ℳ, such that(6.2)-z(φ*f)′(z)(φ*f)(z)≺1+Az1+Bz.
A function f∈ℳ belongs to the class(6.3)𝒲(φ;α):=𝒲(φ;2α-1,α)(0≤α<1)
if it satisfies the condition(6.4)Re{-z(φ*f)′(z)(φ*f)(z)}>α(z∈𝒟).
The class 𝒲(φ;α) is related to the class of starlike function of order α. In particular, we have the following relationships:(6.5)𝒮*(α)=𝒲(1z(1-z);α),𝒮c(α)=𝒲(1-2zz(1-z)2;α).Letλ be a convex parameter. A function f∈ℳ belongs to the class(6.6)𝒲λ(φ;A,B):=𝒲(λzφ(z)+(λ-1)z2φ′(z),1z;A,B)
if it satisfies the condition(6.7)λz(φ*f)(z)+(λ-1)z2(φ*f)′(z)≺1+Az1+Bz.
The classes 𝒲(φ;A,B),𝒲(φ;α), and 𝒲λ(φ;A,B) generalize well-known important classes, which were investigated in earlier works; see for example [5–10].If we apply the results presented in this paper to the classes discussed above, we can obtain several additional results. Some of these results were obtained in earlier works; see for example [5–10].



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