# Sufficiency Criteria for a Class ofp-Valent Analytic Functions of Complex Order

Authors: Muhammad Arif
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

In the present paper, we consider a subclass ofp-valent analytic functions and obtain certain simple sufficiency criteria by using three different methods for the functions belonging to this class. Many known results appear as special consequences of our work.


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## Body

## 1. Introduction

LetAp(n) be the class of functions f(z) analytic and p-valent in the open unit disk 𝕌={z:|z|<1} and of the form
(1)f(z)=zp+∑k=p+n∞akzk(p∈ℕ).In particular,Ap(1)=Ap, A1(n)=A(n), and A1(1)=A. By 𝒮p*(n,b) and 𝒞p(n,b), n,p∈ℕ and b∈ℂ∖{0}, we mean the subclasses of Ap(n) which are defined, respectively, by
(2)Re{1+1p+b-1(zf′(z)f(z)-p)}>0,(z∈𝕌),Re{1+1p+b-1(zf′′(z)f′(z)-p+1)}>0,(z∈𝕌).Forb=1, p=1, n=1, the previous two classes defined in (2) reduce to the well-known classes of starlike and convex, respectively.For functionsf(z), g(z)∈Ap(n) of the form (1), we define the convolution (Hadamard product) of f(z) and g(z) by
(3)(f⋆g)(z)=zp+∑k=p+n∞akbkzk,(z∈𝕌).Now we define the subclassℳp(n,b;g(z)) of Ap(n) by
(4)Re{1+1p+b-1(z(f(z)⋆g(z))′f(z)⋆g(z)-p)}>0,(z∈𝕌).Sufficient conditions were studied by various authors for different subclasses of analytic and multivalent functions, for some of the related work see [1–8]. The object of the present paper is to obtain sufficient conditions for the subclass ℳp(n,b;g(z)) of Ap(n). We also consider some special cases of our results which lead to various interesting corollaries and relevances of some of these with other known results also being mentioned.We will assume throughout our discussion, unless otherwise stated, thatn∈ℕ, p∈ℕ, b∈ℂ∖{0}.


## 2. Preliminary Results

To obtain our main results, we need the following Lemma's.Lemma 1 (see [9]).
Ifq(z)∈A(n) with n≥1 and satisfies the condition
(5)|q′(z)-1|<n+1(n+1)2+1(z∈𝕌),
then
(6)q(z)∈𝒮*(n,1).Lemma 2 (see [10]).
Ifq(z)∈A(n) satisfing the condition
(7)|argq′(z)|<π2δn(z∈𝕌),
where δn is the unique root of the equation
(8)2tan-1[n(1-δn)]+π(1-2δn)=0,
then
(9)q(z)∈𝒮*(n,1).Lemma 3 (see [11]).
LetΩ be a set in the complex plane ℂ, and suppose that Ψ is a mapping from ℂ2×𝕌 to ℂ which satisfies Ψ(ix,y,z)∉Ω for z∈𝕌 and for all real x,y such that y≤(-n/2)(1+x2). If q(z)=1+cnzn+⋯ is analytic in 𝕌 and Ψ(q(z),zq′(z),z)∈Ω for all z∈𝕌, then Req(z)>0.


## 3. Main Results

Theorem 4.
Iff(z)∈Ap(n) satisfies
(10)|(f(z)*g(z)zp)1/(p+b-1){z(f(z)*g(z))′f(z)*g(z)+b-1}-(p+b-1)(f(z)*g(z)zp)|<n+1(n+1)2+1|p+b-1|(z∈𝕌),
then f(z)∈ℳp(n,b;g(z)).Proof.
Let us set a functionp(z) by
(11)p(z)=z(f(z)*g(z)zp)1/(p+b-1)=z+an+pbn+p(p+b-1)zn+⋯
for f(z)∈Ap(n). Then clearly (11) shows that p(z)∈A(n).
Differentiating (11) logarithmically, we have
(12)p′(z)p(z)=1p+b-1[f(z)*g(z)′f(z)*g(z)-pz]+1z
which gives
(13)|p′(z)-1|=|(f(z)*g(z)zp)1/(p+b-1)1p+b-1×{z(f(z)*g(z))′f(z)*g(z)+b-1}-1|.
Thus using (10), we have
(14)|p′(z)-1|≤n+1(n+1)2+1,(z∈𝕌).
Hence, using Lemma1, we have p(z)∈𝒮*(n,1).
From (12), we can write
(15)zp′(z)p(z)=1p+b-1[z(f(z)*g(z))′f(z)*g(z)-p]+1.
Sincep(z)∈𝒮*(n,1), it implies that Re(zp′(z)/p(z))>0. Therefore, we get
(16)Re{1+1p+b-1(z(f(z)*g(z))′f(z)*g(z)-p)}=Rezp′(z)p(z)>0,
and this implies that f(z)∈ℳp(n,b;g(z)).
Settingn=p=1 and g(z)=z/(1-z) in Theorem 4, we get the following.Corollary 5.
Iff(z)∈A satisfies
(17)|(f(z)z)1/b{zf′(z)f(z)+b-1}-b|<2|b|5(z∈𝕌),
then f(z)∈𝒮*(b), the class of starlike functions of complex order b.Puttingn=p=1 and g(z)=z/(1-z)2 in Theorem 4, we have the following.Corollary 6.
Iff(z)∈A satisfies
(18)|(f′(z))(1-b)/b{zf′′(z)+bf′(z)}-b|<2|b|5(z∈𝕌),
then f(z)∈𝒞(b), the class of convex functions of complex order b.Remark 7.
If we putb=1-α in Corollaries 5 and 6, we get the results proved by Uyanık et al. [1]. Furthermore, for b=1, we obtain the results studied by Mocanu [2] and Nunokawa et al. [3], respectively. Also if we set b=1-α with g(z)=zp/(1-zn) and g(z)=zp/(1-zn)2p in Theorem 4, we obtain the results due to Goyal et al. [4].Theorem 8.
Iff(z)∈Ap(n) satisfies
(19)|arg(f(z)*g(z)zp)1/(p+b-1)+arg{1p+b-1(z(f(z)*g(z))′f(z)*g(z)+b-1)}|<π2δn(z∈𝕌),
where δn is the unique root of (8), then f(z)∈ℳp(n,b;g(z)).Proof.
Letp(z) be given by (11), which clearly belongs to the class A(n).
Now differentiating (11), we have
(20)p′(z)=(f(z)*g(z)zp)1/(p+b-1)×1p+b-1{z(f(z)*g(z))′f(z)*g(z)+b-1}
which gives
(21)|argp′(z)|=|arg(f(z)*g(z)zp)1/(p+b-1)+arg{1p+b-1(z(f(z)*g(z))′f(z)*g(z)+b-1)}|.
Thus using (19), we have
(22)|argp′(z)|≤π2δn(z∈𝕌),
where δn is the root of (8). Hence, using Lemma 2, we have p(z)∈𝒮*(n,1).
From (20), we can write
(23)zp′(z)p(z)=1p+b-1[z(f(z)*g(z))′f(z)*g(z)-p]+1.
Sincep(z)∈𝒮*(n,1), it implies that Re(zp′(z)/p(z))>0. Therefore, we get (16), and hence f(z)∈ℳp(n,b;g(z)).
Makingn=1, b=1-α with 0≤α<p and g(z)=zp/(1-z), we have the following.Corollary 9.
Iff(z)∈Ap satisfies
(24)|arg(f(z)zp)+(p-α)arg{zf′(z)f(z)-α}|<π2δ1(p-α)(z∈𝕌),
where δ1 is the unique root of (8) with n=1, then f(z)∈𝒮p*(α), the class of p-valent starlike functions of order α.Also if we taken=1, b=1-α with 0≤α<p and g(z)=zp/(1-z)2p in Theorem 8, we obtain the following result.Corollary 10.
Iff(z)∈Ap satisfies
(25)|arg(f′(z)pzp-1)+(p-α)arg{zf′′(z)f′(z)+1-α}|<π2δ1(p-α)(z∈𝕌),
where δ1 is the unique root of (8) with n=1, then f(z)∈𝒞p(α), the class of p-valent convex functions of order α.Remark 11.
For puttingp=1,α=0 in Corollary 10 and p=1 in Corollary 9, we obtain the results proved by Mocanu [10] and Uyanık et al. [1], respectively.Theorem 12.
Iff(z)∈Ap(n) satisfies
(26)Re[1p+b-1{z(f(z)*g(z))′f(z)*g(z)(ρz(f(z)*g(z))′′(f(z)*g(z))′+1)}+b-11p+b-1]>M24L+N,
where 0≤ρ≤1 and
(27)L=ρ(p+Reb-1+n2),M=2ρImb,N=ρ((-n2)((Reb)2-(Imb)2-Reb)×(p+Reb-1)+(Imb)2(2Reb-1)×((p+Reb-1)2+(Imb)2)-1)-n2),
thenf(z)∈ℳp(n,b;g(z)).Proof.
Let us set(28)z(f(z)*g(z))′f(z)*g(z)=(p+b-1)p(z)-b+1.
Thenp(z) is analytic in 𝕌 with p(0)=1.
Taking logarithmic differentiation of (28) and then by simple computation, we obtain
(29)1p+b-1{z(f(z)*g(z))′f(z)*g(z)×(ρz(f(z)*g(z))′′(f(z)*g(z))′+1)+b-1}=Azp′(z)+Bp2(z)+Cp(z)+D=Ψ(p(z),zp′(z),z)
with
(30)A=ρ,B=ρ(p+b-1),C=-2ρb+ρ+1,D=ρb2-ρbp+b-1.
Now for all realx and y satisfying y≤-(n/2)(1+x2), we have
(31)Ψ(ix,y,z)=Ay-Bx2+C(ix)+D.
Reputing the values ofA, B, C, D and then taking real part, we obtain
(32)ReΨ(ix,y,z)≤-Lx2+Mx+N=-(Lx-M2L)2+M24L+N<M24L+N,
where L, M, N are given in (27).
LetΩ={w:Rew>(M2/4L)+N}. Then Ψ(h(z),zh′(z),z)∈Ω and Ψ(ix,y,z)∉Ω, for all real x and y satisfying y≤-(n/2)(1+x2), z∈𝕌. Using Lemma 3, we have Rep(z)>0. This implies that
(33)Re{1+1p+b-1(z(f(z)*g(z))′f(z)*g(z)-p)}>0,
and hence f(z)∈ℳp(n,b;g(z)).
If we putp=n=1,b=1-α and g(z)=z/(1-z)2(1-α) in Theorem 12, we obtain the following result proved in [12].Corollary 13.
Iff(z)∈A satisfies
(34)Re{zf′(z)f(z)(ρzf′′(z)f′(z)+1)}>αρ(α-12)+(α-ρ2),
then f(z)∈𝒮*(α).Furthermore, forα=0 in Corollary 13, we have the following result proved in [13].Corollary 14.
Iff(z)∈A satisfies
(35)Re{zf′(z)f(z)(ρzf′′(z)f′(z)+1)}>-ρ2,
then f(z)∈𝒮*.



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*Source: 517296-2013-04-03.xml*