# Sufficient Conditions for Meromorphicallyp-Valent  Starlikeness and Close-to-Convexity

Authors: B. A. Frasin; Tariq Al-Hawary; M. Darus
Journal: Geometry
 (2013)

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## Abstract

Making use of the linear operatorIλm defined by (Frasin 2012), we introduce the class 𝕄p,jm(λ,μ,α) of meromorphically p-valent functions in the punctured unit disk 𝒰∗. Furthermore, we obtain some sufficient conditions for starlikeness and close-to-convexity for
functions belonging to this class. Several corollaries and consequences of the main results
are also considered.


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## Body

## 1. Introduction and Definitions

LetΣp,j denote the class of functions of the form:
(1)f(z)=1zp+∑n=j∞‍an+p-1zn+p-1,(p,j∈ℕ:={1,2,3,…}),
which are p-valent in the punctured unit disk 𝒰*=𝒰∖{0}={z:z∈ℂ;|z|<1}. A function f(z) in Σp,j is said to be meromorphically p-valent starlike of order α if and only if
(2)Re{-zf′(z)f(z)}>α,(z∈𝒰*),
for some α(0≤α<p). We denote by Σp,j*(α) the class of all meromorphically p-valent starlike of order δ. Further, a function f(z) in Σp,j is said to be meromorphically p-valent convex of order α if and only if
(3)Re{-1-zf′′(z)f′(z)}>α,(z∈𝒰*),
for some α(0≤α<p). We denote by Σp,jk(α) the class of all meromorphically p-valent convex of order δ. A function f(z) belonging to Σp,j is said to be meromorphically p-valent close-to-convex of order α if it satisfies
(4)Re(-f′(z)z-p-1)>α,(z∈𝒰*),
for some α(0≤α<p). We denote by Σp,jc(α) the subclass of Σp,j consisting of functions which are meromorphically p-valent close-to-convex of order α in 𝒰*.Many interesting families of analytic and multivalent functions were considered by earlier authors in Geometric Functions Theory (cf. e.g., [1–4]). Some subclasses of Σp,j=Σ when p=j=1 were considered by (e.g.) Miller [5], Pommerenke [6], Clunie [7], Owa et al. [8], and Royster [9]. Furthermore, several subclasses of Σp,j=Σp when j=1 were studied by (amongst others) Mogra et al. [10], Uralegaddi and Ganigi [11], Cho et al. [12], Aouf [13, 14], and Uralegaddi and Somanatha [15].For a functionf in Σp,j, Frasin [16] introduced and studied the following differential operator:
(5)I0f(z)=f(z),Iλ1f(z)=(1-λ)f(z)+λzf′(z)+λ(p+1)zp,λ≥0,Iλ2f(z)=(1-λ)Iλ1f(z)+λz(Iλ1f(z))′+λ(p+1)zp,
and for m=1,2,3,…(6)Iλmf(z)=(1-λ)Iλm-1f(z)+λz(Iλm-1f(z))′+λ(p+1)zp=1zp+∑n=j∞‍[1+λ(p+n-2)]man+p-1zn+p-1.Note that forλ=p=j=1, we have the operator Imf(z) introduced and studied by Frasin and Darus [17].It easily verified from (6) that
(7)λz(Iλmf(z))′=Iλm+1f(z)-(1-λ)Iλmf(z)-λ(p+1)zp,λz(Iλmf(z))′′=(Iλm+1f(z))′-(Iλmf(z))′+λp(p+1)zp+1.
Making use of the above operator Iλm, we now introduce a new class of meromorphically and p-valent functions defined as follows.Definition 1.
A functionf(z)∈Σp,j is said to be a member of the class 𝕄p,jm(λ,μ,α) if and only if
(8)|zp+1(Iλmf(z))′(zpIλmf(z))μ-1+p|<p-α,
for some α(0≤α<p);μ≥0;λ≥0,p∈ℕ,m∈ℕ0=ℕ∪{0} and for all z∈𝒰*.Note that condition (8) implies that
(9)Re(-zp+1(Iλmf(z))′(zpIλmf(z))μ-1)>α.Clearly, we have𝕄p,j0(1,2,α)=Σp,j*(α) and 𝕄p,j0(1,1,α)=Σp,jc(α).In this paper, we obtain some sufficient conditions for functions belonging to the class𝕄p,jm(λ,μ,α). Several corollaries and consequences of the main results are also considered.In order to derive our main results, we have to recall the following lemmas.Lemma 2 (see [18]).
Letw(z) be analytic in 𝒰 and such that w(0)=0. Then if |w(z)| attains its maximum value on circle |z|=r<1 at a point  zo∈𝒰, we have(10)zow′(zo)=kw(zo),
where k≥1 is a real number.Lemma 3 (see [19]).
LetΩ be a set in the complex plane ℂ and suppose that Φ(z) is a mapping from ℂ2 × 𝒰 to ℂ which satisfies Φ(ix,y;z)∉Ω for z∈𝒰, and for all real x,y such that y≤-n(1+x22)/2. If the function q(z)=1+qnzn+qn+1zn+1+⋯ is analytic in 𝒰 such that Φ(q(z),zq′(z);z)∈Ωfor all z∈𝒰, then Re{q(z)}>0.Lemma 4 (see [20]).
Letq(z) be analytic in 𝒰 with q(0)=1. If there exists a point z0∈𝒰 such that
(11)Re{q(z)}>0,(|z|<|z0|),Re{q(z0)}=0,q(z)≠0,
then
(12)q(z0)=ia,zq′(z0)q(z0)=ik2(a+1a),
where a∈ℝ∖{0} and k≥1.


## 2. Sufficient Conditions for Meromorphicallyp-Valent Starlikeness and Close-to-Convexity

Making use of Lemma2, we first proveTheorem 5.
Iff(z)∈Σp,j satisfies
(13)|p+1+1λ[(Iλm+1f(z))′(Iλmf(z))′-1+λp(p+1)zp+1(Iλmf(z))′]-(μ-1)[p+1λ(Iλm+1f(z)Iλmf(z)-(1-λ)-λ(p+1)zpIλmf(z))]-γ(zp+1(Iλmf(z))′(zpIλmf(z))μ-1+p)|<(p-α)(γ(2p-α)+1)2p-α,(z∈𝒰),
for some α  (0≤α<p);μ,γ≥0;λ>0,p∈ℕ, and m∈ℕ0, then f(z)∈𝕄p,jm(λ,μ,α).Proof.
Define the functionw(z) by
(14)zp+1(Iλmf(z))′(zpIλmf(z))μ-1=-p+(α-p)w(z).
Then w(z) is analytic in 𝒰 and w(0)=0. It follows from (14) and the identities (7) that
(15)p+1+1λ[(Iλm+1f(z))′(Iλmf(z))′-1+λp(p+1)zp+1(Iλmf(z))′]-(μ-1)[p+1λ(Iλm+1f(z)Iλmf(z)-(1-λ)-λ(p+1)zpIλmf(z))]-γ(zp+1(Iλmf(z))′(zpIλmf(z))μ-1+p)=γ(p-α)w(z)+(p-α)zw′(z)p+(p-α)w(z).
Suppose that there exists z0∈𝒰 such that
(16)max|z|<z0|w(z)|=|w(z0)|=1,
then from Lemma 2, we have (10). Therefore, letting zow′(zo)=keiθ(0≤θ<2π), with k≥1, we obtain that
(17)|p+1+1λ[(Iλm+1f(z0))′(Iλmf(z0))′-1+λp(p+1)z0p+1(Iλmf(z0))′]-(μ-1)[p+1λ(Iλm+1f(z0)Iλmf(z0)-(1-λ)-λ(p+1)z0pIλmf(z0))]-γ(z0p+1(Iλmf(z0))′(z0pIλmf(z0))μ-1+p)|=|γ(p-α)w(z0)+(p-α)zw′(z0)p+(p-α)w(z0)|≥Re{γ(p-α)+(p-α)kp+(p-α)w(z0)}>γ(p-α)+p-α2p-α=(p-α)(γ(2p-α)+1)2p-α,
which contradicts our assumption (13). Therefore we have |w(z)|<1 in 𝒰. Finally, we have
(18)|zp+1(Iλmf(z))′(zpIλmf(z))μ-1+p|=(p-α)|w(z)|<p-α,(z∈𝒰),
that is, f(z)∈𝕄p,jm(λ,μ,α). This completes the proof of the theorem.Next we prove the following.Theorem 6.
Iff(z)∈Σp,j satisfies
(19)Re{(zp+1(Iλmf(z))′(zpIλmf(z))μ-1)2-zp+1(Iλmf(z))′(zpIλmf(z))μ-1×(1+1λ[(Iλm+1f(z))′(Iλmf(z))′-1+λp(p+1)zp+1(Iλmf(z))′]+1-μλ[Iλm+1f(z)Iλmf(z)-λ(p+1)zpIλmf(z)](Iλm+1f(z))′(Iλmf(z))′)(zp+1(Iλmf(z))′(zpIλmf(z))μ-1)2}>δ(δ+(1-μ)(1-λ)λ+n2)+p(δ(μ-2)-n2),
for some δ(0≤δ<p);μ≥0;λ>0,p∈ℕ and m∈ℕ0,  then f(z)∈𝕄p,jm(λ,μ,δ).Proof.
Define the functionq(z) by
(20)zp+1(Iλmf(z))′(zpIλmf(z))μ-1=-δ+(δ-p)q(z).
Then, we see that q(z)=1+qnzn+qn+1zn+1+⋯ is analytic in 𝒰. Differentiating both sides of (20) with respect z logarithmically, we get
(21)1+z(Iλmf(z))′′(Iλmf(z))′+(1-μ)z(Iλmf(z))′Iλmf(z)=(p-δ)zq′(z)δ+(p-δ)q(z)+p(μ-2).
Using the identities (7) in (21), we find that
(22)1+1λ[(Iλm+1f(z))′(Iλmf(z))′-1+λp(p+1)zp+1(Iλmf(z))′]+1-μλ[Iλm+1f(z)Iλmf(z)-λ(p+1)zpIλmf(z)]=(p-δ)zq′(z)δ+(p-δ)q(z)+p(μ-2)+(1-μ)(1-λ)λ.
From (20) and (22), we immediately get
(23)(zp+1(Iλmf(z))′(zpIλmf(z))μ-1)2-zp+1(Iλmf(z))′(zpIλmf(z))μ-1×(1+1λ[(Iλm+1f(z))′(Iλmf(z))′-1+λp(p+1)zp+1(Iλmf(z))′]×+1-μλ[Iλm+1f(z)Iλmf(z)-λ(p+1)zpIλmf(z)](Iλm+1f(z))′(Iλmf(z))′)=(p-δ)zq′(z)+(p-δ)2q2(z)+(p-δ)q(z)×(p(μ-2)+(1-μ)(1-λ)λ+2δ)+(p(μ-2)+(1-μ)(1-λ)λ)δ+δ2=Φ(q(z),zq′(z);z),
where
(24)Φ(r,s;t)=(p-δ)s+(p-δ)2r2+(p-δ)r(p(μ-2)+(1-μ)(1-λ)λ+2δ)+(p(μ-2)+(1-μ)(1-λ)λ)δ+δ2.
For all real x,y satisfying y≤-n(1+x22)/2, we have
(25)ReΦ(ix,y;z)=(p-δ)y-(p-δ)2x2+(p(μ-2)+(1-μ)(1-λ)λ)δ+δ2≤-n2(p-δ)-(p-δ)[n2+p-δ]x2+(p(μ-2)+(1-μ)(1-λ)λ)δ+δ2≤(p(μ-2)+(1-μ)(1-λ)λ)δ+δ2-n2(p-δ)=δ(δ+(1-μ)(1-λ)λ+n2)+p(δ(μ-2)-n2).
Let Ω={w:Rew>δ(δ+(1-μ)(1-λ)/λ+n/2)+p(δ(μ-2)-n/2)}. Then Φ(q(z),zq′(z);z)∈Ω, and Φ(ix,y;z)∉Ω, for all real x and y≤-n(1+x22)/2, z∈𝒰. By using Lemma 3, we have Req(z)>0, that is, f(z)∈𝕄p,jm(λ,μ,δ).Finally, we prove the next theorem.Theorem 7.
Iff(z)∈Σp,j satisfies
(26)Re{(Iλm+1f(z))′(Iλmf(z))′(μ-1)(Iλm+1f(z)Iλmf(z)-λ(p+1)zpIλmf(z))-((Iλm+1f(z))′(Iλmf(z))′-1+λp(p+1)zp+1(Iλmf(z))′)}<λ-pλ(μ-2)-(1-μ)(1-λ)+λ(p-δ)2δ
for some δ  (p/2≤δ<p);μ≥0;λ>0,p∈ℕ,  and m∈ℕ0,  then f(z)∈𝕄p,jm(λ,μ,δ).Proof.
Define the functionq(z) by
(27)-zp+1(Iλmf(z))′(zpIλmf(z))μ-1=δ+(p-δ)q(z).
Then, we see that q(z) is analytic in 𝒰 with q(0)=1. From (22) it follows that and
(28)(μ-1)(Iλm+1f(z)Iλmf(z)-λ(p+1)zpIλmf(z))-((Iλm+1f(z))′(Iλmf(z))′-1+λp(p+1)zp+1(Iλmf(z))′)=λ-pλ(μ-2)-(1-μ)(1-λ)-λ(p-δ)zq′(z)δ+(p-δ)q(z).
If there exists a point z0∈𝒰 such that
(29)Re{q(z)}>0,(|z|<|z0|),Re{q(z0)}=0,q(z)≠0.
Then applying Lemma 4, we have
(30)q(z0)=ia,zq′(z0)q(z0)=ik2(a+1a),
where a∈ℝ∖{0} and k≥1. Thus, from (28) and (30) we get
(31)(μ-1)(Iλm+1f(z0)Iλmf(z0)-λ(p+1)z0pIλmf(z0))-((Iλm+1f(z0))′(Iλmf(z0))′-1+λp(p+1)z0p+1(Iλmf(z0))′)=λ-pλ(μ-2)-(1-μ)(1-λ)-λ(p-δ)z0q′(z0)δ+(p-δ)q(z0)=λ-pλ(μ-2)-(1-μ)(1-λ)+kλ(p-δ)(1+a2)2(δ+i(p-δ)a).
Therefore, we have(32)Re{(Iλm+1f(z0))′(Iλmf(z0))′(μ-1)(Iλm+1f(z0)Iλmf(z0)-λ(p+1)z0pIλmf(z0))-((Iλm+1f(z0))′(Iλmf(z0))′-1+λp(p+1)z0p+1(Iλmf(z0))′)}=λ-pλ(μ-2)-(1-μ)(1-λ)+kλ(p-δ)(1+a2)δ2(δ2+(p-δ)2a2)≥λ-pλ(μ-2)-(1-μ)(1-λ)+kλ(p-δ)2δ≥λ-pλ(μ-2)-(1-μ)(1-λ)+λ(p-δ)2δ.
This contradicts our assumption. Thus, we conclude thatReq(z)>0 for all z∈𝒰, that is,
(33)Re(-zp+1(Iλmf(z))′(zpIλmf(z))μ-1)>δ.


## 3. Special Cases and Consequences

Among the various interesting and important consequences of Theorems5–7, we mention now some of the corollaries relating to the classes Σp,j*(α), and Σp,jc(δ), which are deducible from the main results.Firstly, if we letm=0,  μ=2, and λ=1 in Theorems 5–7, we get the following sufficient conditions for meromorphically p-valent starlike functions.Corollary 8.
Iff(z)∈Σp,j satisfies
(34)|1+zf′′(z)f′(z)-(1+γ)zf′(z)f(z)-γp|<(p-α)(γ(2p-α)+1)2p-α,
for some α(0≤α<p);p∈ℕ and γ≥0,then f(z)∈Σp,j*(α).Corollary 9.
Iff(z)∈Σp,j satisfies
(35)Re{(zf′(z)f(z))(2zf′(z)f(z)-1-zf′′(z)f′(z))}>δ(δ+n2)-np2,
for some δ(0≤δ<p);p∈ℕ and γ≥0, then f(z)∈Σp,j*(δ).Corollary 10.
Iff(z)∈Σp,j satisfies
(36)Re{zf′(z)f(z)-zf′′(z)f′(z)}<1+p-δ2δ,
for some δ(p/2≤δ<p);p∈ℕ,  then f(z)∈Σp,j*(δ).Settingm=0 and μ=λ=1 in Theorems 5–7, we get the following sufficient conditions for meromorphically p-valent close-to-convex functions.Corollary 11.
Iff(z)∈Σp,j satisfies
(37)|p+1+zf′′(z)f′(z)-γ(zp+1f′(z)+p)|<(p-α)(γ(2p-α)+1)2p-α,
for some α(0≤α<p);p∈ℕ and γ≥0,  then f(z)∈Σp,jc(α).Corollary 12.
Iff(z)∈Σp,j satisfies
(38)Re{(zp+1f′(z))(zp+1f′(z)-1-zf′′(z)f′(z))}>(δ-p)(δ+n2),
for some δ(0≤δ<p);p∈ℕ, then f(z)∈Σp,jc(δ).Corollary 13.
Iff(z)∈Σp,j satisfies
(39)Re{-zf′′(z)f′(z)}<1+p+p-δ2δ,
for some δ(p/2≤δ<p);p∈ℕ, then f(z)∈Σp,jc(δ).Settingp=j=1 in Corollary 10, we haveCorollary 14.
Iff(z)∈Σsatisfies
(40)Re{zf′(z)f(z)-zf′′(z)f′(z)}<1+1-δ2δ,
for some δ(1/2≤δ<1), then f(z)∈Σ*(δ). In particular, if f(z)∈Σ satisfies
(41)Re{zf′(z)f(z)-zf′′(z)f′(z)}<32,
then f(z) is meromorphically starlike of order 1/2.Settingp=j=1 in Corollary 13, we have the following.Corollary 15.
Iff(z)∈Σsatisfies
(42)Re{-zf′′(z)f′(z)}<2+1-δ2δ,
for some δ(1/2≤δ<1), then f(z)∈Σc(δ). In particular, if f(z)∈Σ satisfies
(43)Re{-zf′′(z)f′(z)}<52,
then f(z) is meromorphically close-to-convex of order 1/2.Remark 16.
(i) If we putγ=p=j=1 in Corollaries 8 and 11, we get Corollaries 5 and 1, respectively, proved by Goyal and Prajapat [21].
(ii) If we putp=j=n=1 and δ=0 in Corollaries 9 and 12, we get Corollaries 8 and 4, respectively, proved by Goyal and Prajapat [21].



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