# Starlikeness of General Integral Operator for Meromorphic Multivalent Functions

Authors: Pranay Goswami; Serap Bulut
Journal: Journal of Complex Analysis
 (2013)

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## Abstract

We introduce a new integral operatorℐγ,δp(f1,…,fn) for meromorphic multivalent functions. The starlikeness condition of this integral operator is determined. Several special cases are also discussed in the form of corollaries.


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## Body

## 1. Introduction

LetΣp denote the class of all meromorphic functions of the form
(1)f(z)=1zp+∑k=0∞akzk(p∈ℕ:={1,2,…}),
which are analytic and p-valent in the punctured open unit disk
(2)𝕌*={z∈ℂ:0<|z|<1}=𝕌∖{0},
where 𝕌 is the open unit disk 𝕌={z∈ℂ:|z|<1}. In particular, we set Σ1=Σ.A functionf∈Σp is said to be  meromorphic p-valent starlike and belongs to the class ℳ𝒮p*, if it satisfies the inequality:
(3)-ℜ{zf′(z)f(z)}>0.A functionf∈Σp is said to be meromorphic p-valent convex and belongs to the class ℳ𝒞p, if it satisfies the inequality:
(4)-ℜ{1+zf′′(z)f′(z)}>0.We note that(5)f∈ℳ𝒞p⟺-zf′p∈ℳ𝒮p*.
In particular, we set
(6)ℳ𝒮1*=ℳ𝒮*,ℳ𝒞1=ℳ𝒞
for the classes of meromorphic starlike and convex functions, respectively.Analogous to the integral operator defined by Frasin [1] on normalized analytic functions, we now define the following general integral operator for the meromorphic p-valent functions in the class Σp.Definition 1.
Letn∈ℕ, γj,δj∈ℝ+∪{0} for all j=1,2,…,n and γ=(γ1,…,γn),δ=(δ1,…,δn). We introduce a new general integral operator
(7)ℐγ,δp(f1,…,fn):Σpn→Σp,ℐγ,δp(f1,…,fn)(z)=1zp+1∫0z∏j=1n(-up+1pfj′(u))γj×(upfj(u))δjdu(z∈𝕌*).Remark 2.
The integral operator introduced here generalizes the following integral operators.(i)
Ifp=1, then we have the integral operator
(8)ℐγ,δ(f1,…,fn)(z)=1z2∫0z∏j=1n(-u2fj′(u))γj×(ufj(u))δjdu
introduced and studied by Bulut and Goswami [2].(ii)
Ifδ=(0,…,0), then it reduces to the integral operator
(9)𝒥γp(f1,…,fn)(z)=1zp+1∫0z∏j=1n(-up+1pfj′(u))γjdu.(iii)
Ifγ=(0,…,0), then it reduces to the integral operator
(10)𝒢δp(f1,…,fn)(z)=1zp+1∫0z∏j=1n(upfj(u))δjdu.(iv)
Forn=1, γ1=γ, δ1=δ, and f1=f, it reduces to the new integral operator
(11)ℐγ,δp(f)(z)=1zp+1∫0z(-up+1pf′(u))γ(upf(u))δdu.The integral operators defined in (ii) and (iii) are introduced and studied by Mohammed and Darus [3] (also the case p=1 in (ii) and (iii) is defined in [4] and [12], resp.). Furthermore, the case p=1 for the integral operator in (iv) is defined by Bulut and Goswami [2]. For the recent developments on meromorphic functions, refer to [5–10].For the starlikeness of the integral operatorℐγ,δp(f1,…,fn) defined in Definition 1, we need to use the following lemma.Lemma 3 (see [11]).
Letψ:ℂ2→ℂ satisfy the following condition:
(12)ℜ{ψ(is,t)}≤0,(s,t∈ℝ;t≤-|a+is|22).
If the function h(z)=a+h1z+h2z2+⋯, where ℜ(a)>0, is analytic in 𝕌 and
(13)ℜ{ψ(h(z),zh′(z))}>0(z∈𝕌),
then
(14)ℜ{h(z)}>0(z∈𝕌).


## 2. Starlikeness of the Integral Operatorℐγ,δp(f1,…,fn)

In this section, we investigate sufficient conditions for the meromorphically starlikeness of the integral operatorℐγ,δp(f1,…,fn) which is defined in Definition 1.Theorem 4.
Letfj∈Σp,γj,δj∈ℝ+∪{0}  for all j=1,2,…,n and γ=(γ1,…,γn),δ=(δ1,…,δn). If
(15)ℜ{-γjzfj′′(z)fj′(z)-δjzfj′(z)fj(z)}>-pn+(p+1)γj+pδj
for all j=1,2,…,n, then the general integral operator ℐγ,δp(f1,…,fn) defined in Definition 1 belongs to the meromorphic starlike function class ℳ𝒮p*.Proof.
For the sake of simplicity, in the proof, we shall writeℱ(z) instead of ℐγ,δp(f1,…,fn)(z). From (7), it is easy to see that
(16)zp+1ℱ′(z)+(p+1)zpℱ(z)=∏j=1n(-zp+1fj′(z)p)γj(zpfj(z))δj.
Differentiating the above equality with respect to z, we obtain
(17)zp+1ℱ′′(z)+2(p+1)zpℱ′(z)+p(p+1)zp-1ℱ(z)=∑j=1n‍{γj(-zp+1fj′(z)p)γj(zpfj(z))δj×(p+1z+fj′′(z)fj′(z))+δj(-zp+1fj′(z)p)γj×(zpfj(z))δj(pz+fj′(z)fj(z))}×∏s=1(s≠j)n‍(-zp+1fs′(z)p)γs(zpfs(z))δs.
From (16) and (17), we get
(18)zp+1ℱ′′(z)+2(p+1)zpℱ′(z)+p(p+1)zp-1ℱ(z)zp+1ℱ′(z)+(p+1)zpℱ(z)=∑j=1n‍{γj(p+1z+fj′′(z)fj′(z))+δj(pz+fj′(z)fj(z))}
or equivalently
(19)z2ℱ′′(z)+2(p+1)zℱ′(z)+p(p+1)ℱ(z)zℱ′(z)+(p+1)ℱ(z)=∑j=1n‍{γj(p+1+zfj′′(z)fj′(z))+δj(p+zfj′(z)fj(z))}.
After some calculations, we have
(20)-z2ℱ′′(z)+(p+2)zℱ′(z)zℱ′(z)+(p+1)ℱ(z)=p+∑j=1n{γj(-p-1-zfj′′(z)fj′(z))as+δj(-p-zfj′(z)fj(z))}.
We can write the left-hand side of (20) as the following:
(21)-(zℱ′(z)/ℱ(z))((zℱ′′(z)/ℱ′(z))+p+2)(zℱ′(z)/ℱ(z))+p+1=p+∑j=1n{γj(-p-1-zfj′′(z)fj′(z))a+δj(-p-zfj′(z)fj(z))}.
Now we define a regular function h(z) by
(22)h(z)=-zℱ′(z)ℱ(z),
and h(0)=p. Differentiating (22) logarithmically, we obtain
(23)-h(z)+zh′(z)h(z)=1+zℱ′′(z)ℱ′(z).
From (21), (22), and (23), we have
(24)h(z)+zh′(z)-h(z)+p+1=p+∑j=1n‍{γj(-p-1-zfj′′(z)fj′(z))+δj(-p-zfj′(z)fj(z))}.
Let us put
(25)ψ(u,v)=u+v-u+p+1.
From (15), (24) and (25), we get
(26)ℜ{ψ(h(z),zh′(z))}=p+∑j=1n{γj(-p-1-ℜ{zfj′′(z)fj′(z)})+δj(-p-ℜ{zfj′(z)fj(z)})}=p-∑j=1n((p+1)γj+pδj)+∑j=1nℜ{-γjzfj′′(z)fj′(z)-δjzfj′(z)fj(z)}>p-∑j=1n((p+1)γj+pδj)+∑j=1n(-pn+(p+1)γj+pδj)=0.
Now we have to show that
(27)ℜ{ψ(is,t)}≤0,(s,t∈ℝ;t≤-p2+s22).
From (25), we have
(28)ℜ{ψ(is,t)}=ℜ{is+t-is+p+1}=t(p+1)s2+(p+1)2≤-(p2+s2)(p+1)2(s2+(p+1)2)<0.
Thus using Lemma 3, and from (26) and (28), we conclude that ℜ{h(z)}>0, and so
(29)-ℜ{zℱ′(z)ℱ(z)}>0,
that is, the integral operator ℱ=ℐγ,δp(f1,…,fn) is a meromorphic multivalent starlike function.Puttingp=1 in Theorem 4, we get the following Corollary.Corollary 5 (see [2, Theorem 2.1]).
Letfj∈Σ,γj,δj∈ℝ+∪{0} for all j=1,2,…,n and γ=(γ1,…,γn),δ=(δ1,…,δn). If
(30)ℜ{-γjzfj′′(z)fj′(z)-δjzfj′(z)fj(z)}>-1n+2γj+δj
for all j=1,2,…,n, then the general integral operator ℐγ,δ(f1,…,fn) defined in (8) belongs to the meromorphically starlike function class ℳ𝒮*.Takingδ=(0,…,0) in Theorem 4, we get the following.Corollary 6.
Letfj∈Σp,γj>0 for all j=1,2,…,n and γ=(γ1,…,γn). If
(31)ℜ{-γjzfj′′(z)fj′(z)}>-pn+(p+1)γj
for all j=1,2,…,n, then the general integral operator 𝒥γp(f1,…,fn) defined in (9) belongs to the meromorphically starlike function class ℳ𝒮p*.Remark 7.
If we setp=1 in Corollary 6, then we have [12, Theorem 2.1].Next, puttingγ=(0,…,0) in Theorem 4, we get the following.Corollary 8.
Letfj∈Σp, δj>0 for all j=1,2,…,n and δ=(δ1,…,δn). If
(32)ℜ{-δjzfj′(z)fj(z)}>-pn+pδj
for all j=1,2,…,n, then the general integral operator 𝒢δp(f1,…,fn) defined in (10) belongs to the meromorphically starlike function class ℳ𝒮p*.Further takingn=1, γ1=γ, δ1=δ, and f1=f in Theorem 4, we have the following.Corollary 9.
Letf∈Σp and γ,δ>0. If
(33)ℜ{-γzf′′(z)f′(z)-δzf′(z)f(z)}>-p+(p+1)γ+pδ,
then the general integral operator ℐγ,δp(f) defined in (11) belongs to the meromorphically starlike function class ℳ𝒮p*.



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*Source: 690584-2013-01-30.xml*