AAAAbstract and Applied Analysis1687-04091085-3375Hindawi Publishing Corporation39103810.1155/2012/391038391038Research ArticleArgument Property for Certain Analytic FunctionsYangQingLiuJin-LinNoorKhalida Inayat1Department of MathematicsYangzhou UniversityYangzhou 225002Chinayzu.edu.cn20128122011201218092011011120112012Copyright © 2012 Qing Yang and Jin-Lin Liu.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Let P be the class of functions p(z) of the form p(z)=1+n=1cnzn which are analytic in the open unit disk U={z:|z|<1}. The object of the present paper is to derive certain argument inequalities of analytic functions p(z) in P.

1. Introduction

Let P be the class of functions p(z) of the form p(z)=1+n=1cnzn, which are analytic in the open unit disk U={z:|z|<1}. For functions p and g in the class P, we say that p is subordinate to g if there exists an analytic function w in U with w(0)=0, |w(z)|<1  (zU), and such that p(z)=g(w(z))  (zU). We denote this subordination bypg(zU). If g is univalent in U, then this subordination pg is equivalent to p(0)=g(0) and p(U)g(U).

Recently, several authors investigated various argument properties of analytic functions (see, e.g., ). The object of the present paper is to discuss some argument inequalities for p in the class P.

Throughout this paper, we let0<α11,0<α21,β=α1-α2α1+α2,c=eβπi.

In order to prove our main result, we well need the following lemma.

Lemma 1.1 (see [<xref ref-type="bibr" rid="B6">6</xref>]).

Let λ0,λ,a,bR and μC. Also let λ0a0,λ(b+2)0,  (b+1)Re(μ)0,    |b+1|2α1+α2,|a-b-1|1max{α1,α2}. If qP satisfies λ0(q(z))a+λ(q(z))b+2+μ(q(z))b+1+zq(z)(q(z))bh(z)(zU), where h(z)=λ0(1+cz1-z)a((α1+α2)/2)+(1+cz1-z)(1/2)(b+1)(α1+α2)×(μ+λ(1+cz1-z)(α1+α2)/2+α1+α22(z1-z+cz1+cz)) is (close to convex) univalent, then -α2π2<arg(q(z))<α1π2(zU). The bounds α1 and α2 in (1.7) are sharp for the function q defined by q(z)=(1+cz1-z)(α1+α2)/2.

Remark 1.2 (see [<xref ref-type="bibr" rid="B6">6</xref>]).

The function q defined by (1.8) is analytic and univalently convex in U and q(U)={w:wC,  -α2π2<  argw<α1π2}.

2. Main Result

Our main theorem is given by the following.

Theorem 2.1.

Let λ0>0,0<a1max{α1,α2},|b+1|2α1+α2,0a-b-11max{α1,α2}. If pP satisfies -γ2π2<arg(λ0(p(z))a+zp(z)(p(z))b)<γ1π2(zU), where γj=γj(a,b,α1,α2)=aαj+2πtan-1(((α1+α2)/2)cos(βπ/2)cos((a-b-1)αjπ/2)2λ0δj(a,b,α1,α2)+((α1+α2)/2)cos(βπ/2)sin((a-b-1)αjπ/2))(j=1,2),δj(a,b,α1,α2)=[(1-((A)cos(βπ/2))2)1/2+(-1)jsin(βπ/2)]1+A2[1-(A)(cos(βπ/2))2+(-1)j(1-((A)cos(βπ/2))2)1/2sin(βπ/2)]A(j=1,2), where 𝒜 denotes (a-b-1)(α1+α2)/2, then -α2π2<arg(p(z))<α1π2(zU). The bounds γ1 and γ2 in (2.2) are the largest numbers such that (2.4) holds true.

Proof.

By taking λ=μ=0 in Lemma 1.1, we find that if pP satisfies λ0(p(z))a+zp(z)(p(z))bh(z)(zU), where h(z)=(1+cz1-z)(b+1)(α1+α2)/2(λ0(1+cz1-z)(a-b-1)(α1+α2)/2+α1+α22(z1-z+cz1+cz)), then (2.4) holds true.

For z=eiθ, z1 and z-1/c, we get z1-z=-12+i2cotθ2,cz1+cz=12+i2tanθ+βπ2,1+cz1-z=1+ei(θ+βπ)1-eiθ=cos((θ+βπ)/2)sin(θ/2)eα1πi/(α1+α2)0.

We consider the following two cases.

(i) If k(θ)=cosθ+βπ2sinθ2>0, then from (2.7), and (2.6), we have h(eiθ)=(cos((θ+βπ)/2)sin(θ/2))(b+1)(α1+α2)/2e(b+1)α1πi/2×(λ0(cos((θ+βπ)/2)sin(θ/2))(a-b-1)(α1+α2)/2e(a-b-1)α1πi/2+i(α1+α2)cos(βπ/2)4k(θ)), and so arg(h(eiθ))=12aα1π+tan-1(((α1+α2)/2)cos(βπ/2)cos((a-b-1)α1π/2)2λ0k1(θ)+((α1+α2)/2)cos(βπ/2)sin((a-b-1)α1π/2)), where λ0>0,  0(a-b-1)(α1+α2)2, eiθ1, eiθ-1/c, k1(θ)=k(θ)(cos((θ+βπ)/2)sin(θ/2))(a-b-1)(α1+α2)/2>0.

We now calculate the maximum value of k1(θ). It is easy to verify that limθ0k1(θ)=limeiθ-1/ck1(θ)=0 and that k1(θ)=-(a-b-1)(α1+α2)4(cos((θ+βπ)/2)sin(θ/2))(a-b-1)(α1+α2)/2-1cos(βπ/2)(sin(θ/2))2k(θ)+12(cos((θ+βπ)/2)sin(θ/2))(a-b-1)(α1+α2)/2cos(θ+βπ2)=12(cos((θ+βπ)/2)sin(θ/2))(a-b-1)(α1+α2)/2.×(cos(θ+βπ2)-(a-b-1)(α1+α2)2cosβπ2).

Set θ1=cos-1((a-b-1)(α1+α2)2cosβπ2)-βπ2, then k1(θ1)=0. Noting that 0(a-b-1)(α1+α2)2,-1<β<1,  |β|π2<cos-1((a-b-1)(α1+α2)2cosβπ2)<π2, we easily have 0<θ1<π,0<θ1+βπ2<π2,0<θ1+βπ2<π2. Hence, k(θ1)>0, and it follows from (2.11) to (2.16) that 0<k1(θ)k1(θ1)=(sinθ12)-(a-b-1)(α1+α2)(cosθ1+βπ2sinθ12)1+A=(1-cosθ12)-A(12(sin(θ1+βπ2)-sinβπ2))1+A=[(1-((A)cos(βπ/2))2)1/2-sin(βπ/2)]1+A2[1-(A)(cos(βπ/2))2-(1-((A)cos(βπ/2))2)1/2sin(βπ/2)]A=δ1(a,b,α1,α2), where 𝒜 denotes (a-b-1)(α1+α2)/2. Thus, by using (2.1), (2.10), and (2.17), we arrive at π>arg(h(eiθ))arg(h(eiθ1))=12aα1π+tan-1(((α1+α2)/2)cos(βπ/2)cos((a-b-1)α1π/2)2λ0δ1(a,b,α1,α2)+((α1+α2)/2)cos(βπ/2)sin((a-b-1)α1π/2))=γ1π2>0.

(ii) If k(θ)<0, then we obtain h(eiθ)=(-cos((θ+βπ)/2)sin(θ/2))(b+1)(α1+α2)/2e-(b+1)α2πi/2×(λ0(-cos((θ+βπ)/2)sin(θ/2))(a-b-1)(α1+α2)/2e-(a-b-1)α2πi/2+i(α1+α2)cos(βπ/2)4k(θ)), which leads to arg(h(eiθ))=-12aα2π-tan-1(((α1+α2)/2)cos(βπ/2)cos((a-b-1)α2π/2)2λ0k2(θ)+((α1+α2)/2)cos(βπ/2)cos((a-b-1)α2π/2)), where λ0>0,  0(a-b-1)(α1+α2)2, eiθ1, eiθ-1/c, k2(θ)=-k(θ)(-cos((θ+βπ)/2)sin(θ/2))(a-b-1)(α1+α2)/2>0.

Now, we have limθ0k2(θ)=limeiθ-1/ck2(θ)=0,k2(θ)=12(-cos((θ+βπ)/2)sin(θ/2))(a-b-1)(α1+α2)/2((a-b-1)(α1+α2)2cosβπ2-cos(-θ-βπ2)). Let θ2=-cos-1((a-b-1)(α1+α2)2cosβπ2)-βπ2, then k2(θ2)=0, θ1+θ2=-βπ, -π<θ2<0,-π2<θ2+βπ2<0,-π2<θ2+βπ2<0. Hence, we deduce that k(θ2)<0 and 0<k2(θ)k2(θ2)=(-sinθ22)-(a-b-1)(α1+α2)(-cosθ2+βπ2sinθ22)1+A=(1-cosθ22)-A(12(sinβπ2-sin(θ2+βπ2)))1+A=[(1-((A)cos(βπ/2))2)1/2+sin(βπ/2)]1+A2[1-(A)(cos(βπ/2))2+(1-((A)cos(βπ/2))2)1/2sin(βπ/2)]A=δ2(a,b,α1,α2), where 𝒜=(a-b-1)(α1+α2)/2. Further, by using (2.1), (2.20), and (2.25), we find that -π<arg(h(eiθ))arg(h(eiθ2))=-12aα2π-tan-1(((α1+α2)/2)cos(βπ/2)cos((a-b-1)α2π/2)2λ0δ2(a,b,α1,α2)+((α1+α2)/2)cos(βπ/2)cos((a-b-1)α2π/2))=-γ2π2<0. In view of h(0)=1>0, we conclude from (2.18) and (2.26) that h(U) properly contains the angular region -γ2π/2<argw<γ1π/2 in the complex w-plane. Therefore, if pP satisfies (2.2), then the subordination relation (2.5) holds true, and thus we arrive at (2.4).

Furthermore, for the function q defined by (1.8), we have -α2π2<arg(q(z))<α1π2(zU),λ0(q(z))a+zq(z)(q(z))b=h(z). Hence, by using (2.18) and (2.25), we see that the bounds γ1 and γ2 in (2.2) are best possible.

Acknowledgment

The authors would like to express sincere thanks to the referees for careful reading and suggestions which helped them to improve the paper.

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