# On a Subclass of Meromorphic Functions Defined by Hilbert Space Operator

Authors: Thomas Rosy; S. Sunil Varma
Journal: Geometry
 (2013)

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## Abstract

In this paper, we define a new operator on the class of meromorphic
functions and define a subclass using Hilbert space operator. Coefficient
estimate, distortion bounds, extreme points, radii of starlikeness,
and convexity are obtained.


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## Body

## 1. Introduction

LetΣ denote the class of meromorphic functions
(1)f(z)=1z+∑n=1∞‍anzn
defined on U={z∈C:0<|z|<1}. For f∈Σ given by (1) and g∈Σ given by
(2)g(z)=1z+∑n=1∞‍bnzn,
the Hadamard product (or convolution) [1] of f and g is defined by
(3)(f*g)(z)=1z+∑n=1∞‍anbnzn,z∈U.
Many subclasses of meromorphic functions have been defined and studied in the past. In particular, the subclasses Ml,mα1(λ,α),  0≤α<1 and 0≤λ<1  [2], and W(α,β),0≤α<1 and 0<β≤1  [3], are considered by researchers. Let H be a complex Hilbert space and L(H) denote the algebra of all bounded linear operators on H. For a complex-valued function f analytic in a domain E of the complex plane containing the spectrum σ(T) of the bounded linear operator T, let f(T) denote the operator on H defined by the Riesz-Dunford integral [4]
(4)f(T)=12πi∫C(zI-T)-1f(z)dz,
where I is the identity operator on H and C is a positively oriented simple closed rectifiable closed contour containing the spectrum σ(T) in the interior domain [5]. The operator f(T) can also be defined by the following series:
(5)f(T)=∑n=0∞‍f(n)(0)n!Tn
which converges in the norm topology. In this paper, we introduce a subclass MP(α,λ,T) of Σ defined using Hilbert space operator and prove a necessary and sufficient condition for the function to belong to this class, the distortion theorem, radius of starlikeness, and convexity. In [6], Atshan and Buti had defined an operator acting on analytic functions in terms of a definite integral. We modify their operator for meromorphic functions as follows.Lemma 1.
Forf∈Σ given by (1), 0≤μ≤1, and 0≤θ≤1, if the operator Sμθ:Σ→Σ is defined by
(6)Sμθf(z)=1(1-μ)θ+1Γ(θ+1)∫0∞tθ+1e-(t/(1-μ))f(zt)dt,
then
(7)Sμθf(z)=1z+∑n=1∞‍L(n,θ,μ)anzn,
where L(n,θ,μ)=(1-μ)n+1Γ(n+θ+2)/Γ(θ+1).Denote byΣP the class of all functions f∈Σ with an≥0.Definition 2.
For0≤λ<1, 0≤α<1, a function f∈ΣP is in the class MP(α,λ,T) if
(8)∥T(Sμθf(T))′-{(λ-1)Sμθf(T)+λT(Sμθf(T))′}∥<∥T(Sμθf(T))′+(1-2α){(λ-1)Sμθf(T)+λT(Sμθf(T))′}∥
for all operators T with ∥T∥<1 and T≠0,  0 being the zero operator on H.


## 2. Coefficient Bounds

Theorem 3.
A meromorphic functionf∈ΣP given by (1) is in the class MP(α,λ,T) if and only if
(9)∑n=1∞‍[n+α-αλ(n+1)]L(n,θ,μ)an≤1-α.
The result is sharp for f(z)=1/z+((1-α)/[n+α-αλ(n+1)]L(n,θ,μ))zn.Proof.
Letf(T)=T-1+∑n=1∞‍anTn. Assume that (9) holds. Then,
(10)∥T(Sμθf(T))′-{(λ-1)Sμθf(T)+λT(Sμθf(T))′}∥-∥T(Sμθf(T))′+(1-2α){(λ-1)Sμθf(T)+λT(Sμθf(T))′}∥=∥∑n=1∞‍[(n+1)-λ(n+1)]L(n,μ,θ)anTn∥-∥2T-1(1-α)-∑n=1∞[n+(1-2α)(λ-1)+λ(1-2α)n]×L(n,μ,θ)anTn2T-1∥≤∑n=1∞‍[n+α-αλ(n+1)]L(n,θ,μ)an-(1-α)≤0,by(9),
and hence f∈MP(α,λ,T).
Conversely, let(11)∥T(Sμθf(T))′-{(λ-1)Sμθf(T)+λT(Sμθf(T))′}∥<∥T(Sμθf(T))′+(1-2α){(λ-1)Sμθf(T)+λT(Sμθf(T))′}∥.
This implies that
(12)∥∑n=1∞[(n+1)(1-λ)]L(n,θ,μ)anTn+1∥≤∥2(1-α)-∑n=1∞[n+(1-2α)(λ-1)+λ(1-2α)n]≤∥∑n=1∞[(1-2α)n]×L(n,θ,μ)anTn+1∑n=1∞∥.
Choose T=eI,  0<e<1. Then, ∑n=1∞‍[(n+1)(1-λ)]L(n,θ,μ)anen/(2e(1-α)-∑n=1∞‍[n+(1-2α)(λ-1)+λ(1-2α)n]L(n,θ·μ)anen)≤1. As e→1, we obtain (9).Corollary 4.
Iff of the form (1) is in MP(α,λ,T), then
(13)an≤1-α[n+α-αλ(n+1)]L(n,θ,μ).
The result is sharp for the function f(z)=1/z+((1-α)/[n+α-αλ(n+1)]L(n,θ,μ))zn.Theorem 5.
Letf0(z)=1/z and fn(z)=1/z+((1-α)/[n+α-αλ(n+1)]L(n,θ,μ))zn,  n≥1,  0≤α<1,  0≤λ<1. Then, f is in the class MP(α,λ,T) if and only if it can be expressed in the form f(z)=∑n=0∞‍μnfn(z), where μn≥0(n=0,1,2,…) and ∑n=0∞‍μn=1.Proof.
Suppose thatf(z)=∑n=0∞‍μnfn(z)=1/z+∑n=1∞‍μn((1-α)/[n+α-αλ(n+1)]L(n,θ,μ))zn; then, we have
(14)∑n=1∞[n+α-αλ(n+1)]L(n,θ,μ)×μn(1-α)[n+α-αλ(n+1)]L(n,θ,μ)=∑n=1∞μn=1-μ0<1.
By Theorem 3, f∈MP(α,λ,T). Conversely, assume that f is in the class MP(α,λ,T); then, by Corollary 4,
(15)an≤1-α[n+α-αλ(n+1)]L(n,θ,μ).
Set μn=([n+α-αλ(n+1)]L(n,θ,μ)/(1-α))an, n=1,2,… and μ0=1-∑n=1∞‍μn. Then, f(z)=∑n=0∞‍μnfn(z)∈MP(α,λ,T).


## 3. Distortion Bounds

In this section, we obtain growth and distortion bounds for the classMP(α,λ,T).Theorem 6.
Iff∈MP(α,λ,T), then
(16)∥f(T)∥≥1∥T∥-1-α(1+α-2αλ)(1-μ)2(θ+1)(θ+2)∥T∥,∥f(T)∥≤1∥T∥+1-α(1+α-2αλ)(1-μ)2(θ+1)(θ+2)∥T∥.
The result is sharp for f(z)=1/z+((1-α)/(1+α-2αλ)(1-μ)2(θ+1)(θ+2))z.Proof.
By Theorem3,
(17)(1-α-2αλ)L(1,θ,μ)∑n=2∞‍an≤∑n=1∞‍[n+α-αλ(n+1)]L(n,θ,μ)an≤1-α.
Therefore,
(18)∑n=1∞‍an≤1-α(1+α-2αλ)L(1,θ,μ)=1-α(1+α-2αλ)(1-μ)2(θ+1)(θ+2).
Also, if f(T)=T-1+∑n=1∞‍anTn, then
(19)1∥T∥-∑n=1∞‍an∥T∥n≤∥f(T)∥≤1∥T∥+∑n=1∞‍an∥T∥n.
Since ∥T∥<1, the above inequality becomes
(20)1∥T∥-∥T∥∑n=1∞‍an≤∥f(T)∥≤1∥T∥+∥T∥∑n=1∞‍an.
Using (18), we get the result.Theorem 7.
Iff∈MP(α,λ,T), then
(21)∥f′(T)∥≥1∥T∥2-1-α(1+α-2αλ)(1-μ)2(θ+1)(θ+2),∥f′(T)∥≤1∥T∥2+1-α(1+α-2αλ)(1-μ)2(θ+1)(θ+2).
The result is sharp for f(z)=1/z+((1-α)/(1+α-2αλ)(1-μ)2(θ+1)(θ+2))z.


## 4. Radii Results

We now find the radius of meromorphically starlikeness and convexity for functionsf in the class MP(α,λ,T).Theorem 8.
Letf∈MP(α,λ,T). Then, f is meromorphically starlike of order δ,  (0≤δ<1) in |z|<r1, where
(22)r1=infn≥1[(1-δ)(n+α-αλ(n+1))L(n,θ,μ)(1-α)(n+2-δ)]1/(n+1),mmmmmmmmmmmmmmmmmmmmmmmn≥1.Proof.
Letf(T)=T-1+∑n=1∞‍anTn. Since f∈MP(α,λ,T) is meromorphically starlike of order δ,
(23)∥-Tf′(T)f(T)-1∥≤1-δ.
Substituting for f, the above inequality becomes
(24)∑n=1∞‍(n+2-δ1-δ)∥T∥n+1an≤1.
By Theorem 3,
(25)∑n=1∞‍[n+α-αλ(n+1)]L(n,θ,μ)1-αan≤1.
Thus, (24) will be true if
(26)(n+2-δ1-δ)∥T∥n+1≤[n+α-αλ(n+1)]L(n,θ,μ)(1-α),
that is,
(27)∥T∥≤[(1-δ)(n+α-αλ(n+1))L(n,θ,μ)(1-α)(n+2-δ)]1/(n+1).Theorem 9.
Letf∈MP(α,λ,T). Then, f is meromorphically convex of order δ,(0≤δ<1) in |z|<r2, where
(28)r2=infn≥1[(1-δ)(n+α-αλ(n+1))L(n,θ,μ)(1-α)n(n+2-δ)]1/(n+1).Theorem 10.
The classMP(α,λ,T) is closed under convex combination.Proof.
Letf(z)=T-1+∑n=0∞‍anTn and g(z)=T-1+∑n=0∞‍bnTn∈MP(α,λ,T). Then, by Theorem 3,
(29)∑n=1∞‍[n+α-αλ(n+1)]L(n,θ,μ)an≤1-α,∑n=1∞‍[n+α-αλ(n+1)]L(n,θ,μ)bn≤1-α.
Define h(T)=λf(T)+(1-λ)g(T). Then, h(T)=T-1+∑n=1∞[λan+(1-λ)bn]Tn. Now,
(30)∑n=1∞‍[n+α-αλ(n+1)]L(n,θ,μ)[λan+(1-λ)bn]=λ∑n=1∞‍[n+α-αλ(n+1)]L(n,θ,μ)an=+(1-λ)∑n=1∞‍[n+α-αλ(n+1)]L(n,θ,μ)bn≤λ(1-α)+(1-λ)(1-α)(using(30))=1-α.
Thus, h∈MP(α,λ,T).


## 5. Hadamard Product

Theorem 11.
The Hadamard productf*g of two functions f and g in MP(α,λ,T) belongs to the class MP(ξ,λ,T), where  ξ=1-2(1-α)2(1-λ)/(1-α)2(1-2λ)+(1-α-2αλ)2(1-μ)2(θ+1)(θ+2).Proof.
It suffices to prove that(31)∑n=1∞‍[n+ξ-ξλ(n+1)]L(n,θ,μ)1-ξanbn≤1,
where
(32)ξ=1-(2(1-α)2(1-λ))×((1-μ)2(θ+1)(θ+2)(1-α)2(1-2λ)+(1-α-2αλ)2mm×(1-μ)2(θ+1)(θ+2))-1.
Since f,g∈MP(α,λ,T)(33)∑n=1∞[n+α-αλ(n+1)]L(n,θ,μ)1-αan≤1,∑n=1∞[n+α-αλ(n+1)]L(n,θ,μ)1-αbn≤1.
By Cauchy-Schwartz inequality,
(34)∑n=1∞[n+α-αλ(n+1)]L(n,θ,μ)1-αanbn≤1.
We have to find the largest ξ such that
(35)[n+ξ-ξλ(n+1)]L(n,θ,μ)1-ξanbn≤[n+α-αλ(n+1)]L(n,θ,μ)1-αanbn,
that is,
(36)anbn≤1-α[n+α-αλ(n+1)]L(n,θ,μ).
It is enough to find the largest ξ such that
(37)1-α[n+α-αλ(n+1)]L(n·θ,μ)≤(1-ξ)[n+α-αλ(n+1)](1-α)[n+ξ-ξλ(n+1)]
which yields
(38)ξ≤((n+α-αλ(n+1))2L(n,θ,μ)-(1-α)2n)×((n+α-αλ(n+1))2L(n,θ,μ)(1-α)2(1-λ(n+1))mmm+(n+α-αλ(n+1))2L(n,θ,μ))-1.
It follows from (38) that
(39)ξ≤1-(1-α)2(n+1)(1-λ)×((n+α-αλ(n+1))2L(n,θ,μ)(1-α)2(1-λ(n+1))mmm+(n+α-αλ(n+1))2L(n,θ,μ))-1.
Let(40)δ(n)=(1-α)2(n+1)(1-λ)×((n+α-αλ(n+1))2L(n,θ,μ)(1-α)2(1-λ(n+1))mmm+(n+α-αλ(n+1))2L(n,θ,μ))-1.
Clearlyδ(n) is an increasing function of n  (n≥1). Letting n=1, we prove the assertion.



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