# On Certain Classes of Meromorphic Functions Associated with Conic Domains

Authors: Khalida Inayat Noor; Fiaz Amber
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

Making use of the concept ofk-uniformly bounded boundary rotation and Ruscheweyh differential operator, we introduce some new classes of meromorphic functions in the punctured unit disc. Convolution technique and principle of subordination are used to investigate these classes. Inclusion results, generalized Bernardi integral operator, and rate of growth of coefficients are studied. Some interesting consequences are also derived from the main results.


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## Body

## 1. Introduction

Let∑ denote the class of functions f of the form (1.1)f(z)=1z+∑n=0∞anzn,
which are analytic in punctured unit disc E*={z:0<|z|<1}=E∖{0}. At z=0, the function f(z) has a simple pole.Let, for0≤β<1, ∑*(β) and ∑c(β) be well-known subclasses of ∑ consisting of functions meromorphic starlike and meromorphic convex of order β, respectively, see [1].Letf,g∈∑, g(z)=1/z+∑n=0∞bnzn and f be given by (1.1). Then convolution (Hadamard product) f*g of f and g is defined by
(1.2)(f*g)(z)=1z+∑n=0∞anbnzn.
Robertson [2] showed that f*g also belongs to ∑.Letf∈∑ and define, for α>-1,z∈E*,Dα:∑→∑ as
(1.3)Dαf(z)=1z(1-z)α+1*f(z).
It can easily be seen that, for α=n∈N0={0,1,2,…},
(1.4)Dnf(z)=z-1n!dndzn(zn+1f(z)).
We note that
(1.5)D0f(z)=f(z),D1f(z)=zf′(z)+2f(z),
and so
(1.6)D1f(z)-2D0f(z)=zf′(z)=2z-1z(1-z)2*f(z).
Equation (1.6) can be verified as follows.Since(1.7)f(z)=1z+∑n=0∞anzn=f(z)*(1z(1-z)),
we have
(1.8)-zf′(z)=1z-∑n=0∞nanzn=f(z)*(1z-z(1-z)2),z∈E*,=f(z)*(1-2zz(1-z)2).
From (1.3), we can readily obtain the following identity for f∈∑ and α>-1:(1.9)z(Dαf(z))′=(α+1)Dα+1f(z)-(α+2)Dαf(z).
Dα, α>1 is known as generalized Ruscheweyh derivative for meromorphic functions.Fork∈[0,1], define the domain Ωk as follows, see [3]:(1.10)Ωk={u+iv:u>k(u-1)2+v2}.
For fixed k, Ωk represents the conic region bounded successively, by the imaginary axis (k=0), the right branch of hyperbola (0<k<1), and a parabola(k=1). Related with Ωk, the domain Ωk,γ can be defined as below, see [4]:(1.11)Ωk,γ=(1-γ)Ωk+γ,(0≤γ<1).
The functions pk,γ(z) with pk,γ(0)=1,pk,γ′(0)>0, univalent in E={z:|z|<1}, map E onto Ωk,γ and are given as in the following:
(1.12)pk,γ(z)={1+(1-2γ)z(1-z),(k=0),1+2(1-γ)π2(log⁡1+z1-z)2,(k=1),1+2(1-γ)1-k2sinh⁡2[(2πarccos⁡k)arctanh⁡z],(0<k<1).The functionspk,γ(z) are continuous as regard to k, have real coefficients for k∈[0,1], and play the part of extremal ones for many problems related to Ωk,γ.LetP be the class of analytic functions with positive real part, and let P(pk,γ)⊂P be the class of functions p(z) which are analytic in E, p(0)=1 such that p(z)≺pk,γ(z) for z∈E, where “≺” denotes subordination, and pk,γ(z) are given by (1.12).We define the following.Definition 1.1.
Letp(z) be analytic in E with p(0)=1. Then p(z) is said to belong to the class Pm(pk,γ), for m≥2,0≤γ<1,k≥0, if and only if there exist p1,p2∈P(pk,γ) such that
(1.13)p(z)=(m4+12)p1(z)-(m4-12)p2(z),z∈E.
We note that(i)
k=0, Pm(p0,γ)=Pm(γ) and with m=2,P2(γ) coincides with P(γ) and p∈P(γ) implies Rep(z)>γ in E;(ii)
whenk=0, γ=0, we have the class Pm introduced in [5].Definition 1.2.
Letf∈∑. Then f(z) is said to belong to the class k-MRm(γ) if and only if (-zf′/f)∈Pm(pk,γ) in E.
Fork=0,m=2, we obtain the class ∑*(γ) of meromorphic starlike functions of order γ.
We can define the classk-MVm(γ) by the following relation:
(1.14)f∈k-MVm(γ)if -zf′∈k-MRm(γ).
When k=0, m=2, γ=0, we obtain ∑c of meromorphic convex functions.Definition 1.3.
Letf∈∑. Then f∈k-MRmα(γ) if and only if Dαf∈k-MRm(γ) for z∈E*.
Similarlyf∈k-MVmα(γ) if and only if Dαf∈k-MVm(γ). We note that the classes k-MRmα(γ) and k-MVmα(γ) are related by relation (1.14).
Fork=0, γ=0, we have 0-MVm(0)=MVm, the class of meromorphic functions of bounded boundary rotation which was studied in [6]. The functions f∈MVm have integral representation of the form
(1.15)f′(z)=-1z2exp⁡{∫02πlog⁡(1-ze-it)dμ(t)},
where μ(t) is a real-valued function of bounded variation on [0,2π] satisfying the conditions
(1.16)∫02πdμ(t)=2,∫02π|dμ(t)|≤m,∫02πe-itdμ(t)=0.With simple computations, it can easily be seen that the third of conditions (1.16) guarantees that the singularity of f(z) at z=0 is a simple pole with no logarithm term.Also it is known thatf∈MVm if and only if f(E) is a domain containing infinity with boundary rotation at most mπ, see [6]. The class Vm is wellknown [1] and consists of analytic functions with boundary rotation at most mπ. Noonan [6] established the relation between the classes Vm and MVm as follows.A functionf∈MVm if and only if there exists g∈Vm of the form g(z)=z+∑n=2∞bnzn with b2=0 such that
(1.17)-1z2f′(z)=g′(z).It is also shown [6] that, for f∈MVm,m>4, there exist ϕ1,ϕ2∈∑* given by ϕ1(z)=1/z+∑n=0∞anzn, ϕ2(z)=1/z+∑n=0∞bnzn, such that a0=((m-2)/(m+2))b0 and
(1.18)f′(z)=-1z2[zϕ1(z)](m+2)/4[zϕ2(z)](m-2)/4.We note thatϕi∈∑* and therefore 1/ϕi∈S* of analytic functions, i=1,2 in (1.18). This give us
(1.19)|zϕ1(z)|≤4,[zϕ2(z)]-1≺(1-z)2,
by distortion results and subordination for the class S*.We can easily extend the relations (1.17) and (1.18) by noting that f∈Vm(γ) implies that there exists f1∈Vm such that f′(z)=(f1′(z))1-γ, z∈E, see [7]. For f∈MVm(γ), m>((2/(1-γ))+2), we can write relation (1.18) as
(1.20)f′(z)=-1z2[zϕ1(z)]((m+2)/4)(1-γ)[zϕ2(z)]((m-2)/4)(1-γ),ϕ1,ϕ2∈∑*.
Throughout this paper, we will assume k∈[0,1], m≥2, γ∈[0,1) and α>1 unless otherwise stated.We also note that all the results proved in this paper hold fork≥0 in general.


## 2. Preliminary Results

The following lemma is a generalized version of a result proved in [3].Lemma 2.1 (see [4]).
Let0≤k<∞ and let β,δ be any complex numbers with β≠0 and Re(βk/(k+1)+δ)>γ. If h(z) is analytic in E,h(0)=1 and satisfies
(2.1){h(z)+zh′(z)βh(z)+δ}≺pk,γ(z),
and qk,γ(z) is an analytic solution of
(2.2){qk,γ(z)+zq′k,γ(z)βqk,γ(z)+δ}=pk,γ(z),
then qk,γ(z) is univalent,
(2.3)h(z)≺qk,γ(z)≺pk,γ(z),
and qk,γ(z) is the best dominant of (2.1).Lemma 2.2 (see [8]).
Letu=u1+iu2, v=v1+iv2 and let ψ(u,v) be complex-valued function satisfying the following conditions:(i)
ψ(u,v) is continuous in a domain D⊂ℂ2,(ii)
(1,0)∈D and ψ(1,0)>0,(iii)
Reψ(iu2,v1)≤0 whenever (iu2,v1)∈D and v1≤-(1+u22)/2.If h(z)=1+∑n=1cnzn is a function analytic in E such that (h(z),zh′(z))∈D, and Re[ψ(h(z),zh′(z))]>0 for z∈E, then Re(h(z))>0 for z∈E.Lemma 2.3 (see [9]).
Letp and q be analytic in E and Rep(z)≥0,q(0)=1 and Req(z)>0 for z∈E. Further let A≠0 and B be complex constants such that A+B≠0. Then
(2.4)Re{p(z)+zp′(z)A+Bq(z)}>0in|z|<ρ(A,B),
where
(2.5)ρ(A,B)=|A+B|[|A|2+2+4|B|+|B|2+R]1/2,R=[|A|2+2+4|B|+|B|2-|A2-B2|2].
This result is sharp for A real and nonnegative constant.


## 3. Main Results

Theorem 3.1.
One has(3.1)k-MRmα+1(γ)⊂k-MRmα,γ=1α+2.
This result is best possible and sharpness follows from the best dominant property.Proof.
Letf∈k-MRmα+1(γ), and set
(3.2)f1(z)=z[zDαf(z)]-1/(1+α),
with
(3.3)r1=sup⁡{r:f1(z)≠0,0<|z|<r<1}.
Then f1(z) is single-valued in 0<|z|<r1.
Using identity (1.9), it follows that
(3.4)p(z)=zf1′(z)f1(z)=-(Dα+1f(z)Dαf(z)-2),
is analytic in |z|<r1 and p(0)=1.
Now, from (1.9) and (3.4), we obtain
(3.5)-(Dα+2f(z)Dα+1f(z)-2)=α+1α+2[p(z)+zp′(z)(α+1)(2-p(z))]+1α+2=α+1α+2[p(z)+zp′(z)2(α+1)-(α+1)p(z)+1α+1].
With h(z)=(1-γ)p(z)+γ, γ=1/(α+2), we can write (3.5) as
(3.6)-(Dα+2f(z)Dα+1f(z)-2)=h(z)+zh′(z)βh(z)+δ,
where β=-(α+2), δ=2α+3.
Sincef∈k-MRmα+1(γ), it follows from (3.6) that
(3.7)(h(z)+(1/β)zh′(z)h(z)+δ/β)∈Pm(pk,γ).
Define
(3.8)ϕδ,β(z)=1z(1-z)1/β+1{1-(β/(β+δ))z1-z},
and let
(3.9)h(z)=(m4+12)h1(z)-(m4-12)h2(z),p(z)=(m4+12)p1(z)-(m4-12)p2(z).
Then using convolution technique, we have
(3.10)(h(z)*ϕδ,β(z)z)=h(z)+zh′(z)βh(z)+δ=(m4+12)(h1(z)+zh1′(z)βh1(z)+δ)-(m4-12)(h2(z)+zh2′(z)βh2(z)+δ).
Thus, from (3.7) and (3.10), we obtain
(3.11)(hi(z)+zh′i(z)βhi(z)+δ)≺pk,γ(z),i=1,2.
It can easily be seen that Re((βk/(k+1))+δ)>γ, so we apply Lemma 2.1 to have from (3.11)
(3.12)hi(z)≺qk,γ(z)≺pk,γ(z),i=1,2,
where qk,γ(z) is the best dominant and is given as
(3.13)qk,γ(z)={[β∫01(tβ+δ-1exp⁡∫0tzpk,γ(u)-1udu)βdt]-1-δβ}.
Since hi(z)=pi(z)*pγ(z), pγ(z)=(1+(1-2γ)z)/(1-z), i=1,2, we have
(3.14)pi(z)*pγ(z)≺pk(z)*pγ(z),
and from this it follows that pi(z)≺pk(z) for i=1,2.
Now from (3.4) we have p∈Pm(pk,0) in E and the proof is complete.As a special case, we have the following.Corollary 3.2.
Letk=0 in Theorem 3.1. Then MRmα+1(γ)⊂MRmα(γ1),where γ=1/(α+2) and
(3.15)γ1=6{(2α+9)+(2α+9)2-24}.Proof.
From (3.6), we have
(3.16)-(Dα+2f(z)Dα+1f(z)-2)=h(z)+zh′(z)βh(z)+δ,β=-(α+2),δ=(2α+3).
Proceeding as in Theorem 3.1, it follows that
(3.17)(hi(z)+zhi′(z)βhi(z)+δ)∈P(γ),i=1,2,z∈E.
Let hi(z)=(1-γ1)Hi(z)+γ1, i=1,2.
Then(3.18){(γ1-γ)+(1-γ1)Hi(z)+(1-γ1)zHi′(z)β(1-γ1)Hi(z)+βγ1+δ}∈P.
We construct a functional ψ(u,v) by taking u=Hi(z), v=zHi′(z). Then
(3.19)ψ(u,v)=(γ1-γ)+(1-γ1)u+(1-γ1)vβ(1-γ1)u+βγ1+δ.
The first two conditions of Lemma 2.2 are easily verified. For condition (iii), we proceed as follows:
(3.20)Reψ(iu2,v1)=(γ1-γ)+(1-γ1)(βγ1+δ)v1(βγ1+δ)2+β2(1-γ1)2u22.
By putting v1≤-(1+u22)/2, we have
(3.21)Reψ(iu2,v1)≤A1+B1u222CiffA1≤0,B1≤0,
where
(3.22)A1=2(γ1-γ)(βγ1+δ)2-(1-γ1)(βγ1+δ),B1=2(γ1-γ)[β2(1-γ1)2]-(1-γ1)(βγ1+δ),C=(βγ1+δ)2+β2(1-γ1)2u22>0.
From A1≤0, we obtain γ1 as given by (3.15) and B1≤0 ensures 0≤γ1<1.
Applying Lemma2.2, we now have Hi∈P, i=1,2 and therefore hi∈P(γ1) in E, consequently h∈Pm(γ1) in E and the proof is complete.We note that, forα=0, we have
(3.23)MRm(12)⊂MRm(69+57).
Also, for α=1, γ=1/3 and γ1≈6/21.We will now investigate the rate of growth of coefficients forf∈k-MVmα(γ) and the corresponding result for the class k-MRmα(γ) will follow from the relation (1.14).Theorem 3.3.
Letf∈k-MVmα(γ) and be given by (1.1). Then, for m>(2/(1-σ)+2), σ=(k+γ)/(1+k), n≥3, one has
(3.24)an=O(1)nβ1-(α+2),β1=(1-σ)(m2-1),
and O(1) depends only on m and σ.
The exponent{β1-(α+2)} in (3.24) is best possible for the class MVmα(σ) as can be seen from the function f0∈MVmα(σ) given by
(3.25)Dαf0(z)=-1z2(1+z2-2z((m-2)/(m+2)))((m+2)/4)(1-σ)(1-z)β1.Proof.
Sincek-MVmα(γ)⊂MVmα(σ), σ=(k+γ)/(1+k) and f∈k-MVmα(γ) implies Dαf∈MVm(σ), we use (1.20) to write
(3.26)F′(z)=(Dαf(z))′=-1z2[zϕ1(z)]((m+2)/4)(1-σ)[zϕ2(z)]((m-2)/4)(1-σ),m>(21-σ+2),
and ϕ1,ϕ2∈∑*.
Now, withF(z)=(1/z)+∑n=0∞Anzn and z=reiθ,0<r<1, we have
(3.27)nAn=12πrn+1∫02πz2F′(z)e-i(n+1)θdθ.
Pommerenke [10] has shown that
(3.28)∫02π1|1-z|λdθ~c(λ)1(1-r)λ-1,(r⟶1)forλ>1.
Thus we use (1.19), (3.26), and (3.28) to have from (3.27)
(3.29)|nAn|≤12πrn+1∫02π|z2F′(z)|dθ≤c(σ,m)(1-γ)(1-σ)((m/2)-1)-1.
We take r=1-(1/n), An=(Γ(n+α+2)/(Γ(α+1)(n+1)!))an, Γ denotes gamma function, and have
(3.30)an=O(1)nβ1-(α+2),
where β1 is as given in (3.24) and O(1) is a constant depending only on m and σ.
This completes the proof.Next we will show that the classk-MRmα(γ) is preserved under an integral operator.ForRec>0, the generalized Bernardi operator for the class ∑ is defined in [11] as below.Letf∈Σ and be given by (1.1). Then the integral transform Fc is defined as
(3.31)F(z)=czc+1∫0ztcf(t)dt=[1z+∑n=0∞cc+n+1anzn]*f(z).
Also
(3.32)Dαf(z)=1z+∑n=0∞Γ(n+α+2)Γ(α+1)(n+1)!anzn.
It easily follows from (3.31) that
(3.33)z(DαF(z))′=cDαf(z)-(1+c)DαF(z),
and as in (1.9),
(3.34)z(DαF(z))′=(α+1)Dα+1F(z)-(α+2)DαF(z).
From (3.33) and (3.34), we have
(3.35)cDαf(z)=(c-α-1)DαF(z)+(α+1)Dα+1F(z).
We now prove the following.Theorem 3.4.
Letf∈k-MRmα(γ). Then F(z), defined by (3.31), also belong to the same class in E*.Proof.
We put(3.36)F1(z)=z[zDαF(z)]-1/(α+1),r1=sup⁡{r:F(z)≠0,0<|z|<1}.
Then F1(z) is single valued and analytic in |z|<r1 and H(z) defined by
(3.37)H(z)={(m4+12)H1(z)-(m4-12)H2(z)}=zF1′(z)F1(z)=-(Dα+1F(z)DαF(z)-2),
is analytic in |z|<r1,H(0)=1.
Form (3.33), (3.34), and (3.37), we obtain
(3.38)-(Dα+1f(z)Dαf(z)-2)={H(z)+zH′(z)(c+α+1)-(α+1)H(z)},|z|<r1.
Since f∈k-MRmα(γ), it follows that
(3.39){H(z)+zH′(z)βH(z)+δ}∈Pm(pk,γ),β=-(α+1),δ=(c+α+1),
and with the convolution technique used before, we have, for i=1,2(3.40){Hi(z)+zHi′(z)βHi(z)+δ}≺pk,γ(z).
Since Re[βk/(k+1)+δ]>γ, we apply Lemma 2.1 to have Hi(z)≺qk,γ(z)≺pk,γ(z), where qk,γ(z) is the best dominant. The required result now follows from (3.37).Corollary 3.5.
Letk=0,γ=0. Then f∈MRmα and F, defined by (3.31), belongs to MRmα(γ2), where γ2 is given as
(3.41)γ2=12(c+α+1).The proof follows on the similar lines of Corollary3.2.Remark 3.6.
We note thatP(pk,γ)⊂P((k+γ)/(1+k)), (0≤γ<1), and therefore Pm(pk,γ)⊂Pm(σ), σ=(k+γ)/(1+k), (0≤σ<1).We prove a partial converse of Theorem3.4 as following.Theorem 3.7.
LetF(z) be defined by (3.31) and let, for Rec>0, F∈MRmα(σ), σ=(k+γ)/(1+k). Then f∈MRmα(σ) for |z|<ρ, where
(3.42)ρ=ρ(A,B)=|A+B|[|A|2+2+4|B|+|B|2+R]1/2,A=c+(1+α)(1-σ),B=-(1+α)(1-σ),R=[(|A|2+2+4|B|+|B|2)2-|A2-B2|2].Proof.
We write(3.43)p(z)=11-σ[-(Dα+1F(z)DαF(z)-2)-σ]=(m4+12)p1(z)-(m4-12)p2(z).
Since F∈MRmα(σ), p∈Pm in E with p(0)=1, and pi∈P, i=1,2.
Proceeding on the similar lines as before, we obtain form (3.42) and (3.43)
(3.44)11-σ[-(Dα+1F(z)DαF(z)-2)-σ]=p(z)+zp′(z){c+(α+1)(1-σ)}=(1+α)(1-σ)p(z)
and with convolution technique as previously used, we get from (3.44)
(3.45)11-σ[-(Dα+1F(z)DαF(z)-2)-σ]=(m4+12)[p1(z)+zp1′(z)A+Bp1(z)]-(m4-12)[p2(z)+zp2′(z)A+Bp2(z)],
where p1, p2∈P, A=c+(1+α)(1-σ), B=-(1+α)(1-σ), (A+B)≠0.
Then, by using, Lemma2.3, we have
(3.46)Re{pi(z)+zpi′(z)A+Bpi(z)}>0in|z|<ρ(A,B),
where ρ(A,B) is given by (3.42).
Now, from (3.44) and (3.46), we have the required result that f∈MRmα(σ) in |z|<ρ(A,B).
From Lemma2.3 it follows that this result is sharp for c>0.



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