AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 346162 10.1155/2014/346162 346162 Research Article On a New Criterion for Meromorphic Starlike Functions Shi Lei http://orcid.org/0000-0001-6118-7196 Wang Zhi-Gang Aouf Mohamed K. School of Mathematics and Statistics Anyang Normal University Anyang Henan 455000 China aynu.edu.cn 2014 1932014 2014 14 12 2013 17 02 2014 19 3 2014 2014 Copyright © 2014 Lei Shi and Zhi-Gang Wang. 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.

The main purpose of this paper is to derive a new criterion for meromorphic starlike functions of order α.

1. Introduction and Preliminaries

Let Σn denote the class of functions of the form (1)f(z)=1z+k=nak-1zk-1(n:={1,2,}), which are analytic in the punctured open unit disk (2)𝕌*:={z:zand  0<|z|<1}=:𝕌{0}.

A function fΣn is said to be in the class 𝒮n*(α) of meromorphic starlike functions of order α if it satisfies the condition (3)(zf(z)f(z))<-α  (z𝕌;0α<1). For simplicity, we write 𝒮n*(0)=:𝒮n*.

For two functions f and g, analytic in 𝕌, we say that the function f is subordinate to g in 𝕌 and write (4)f(z)g(z)(z𝕌), if there exists a Schwarz function ω, which is analytic in 𝕌 with (5)ω(0)=0,|ω(z)|<1(z𝕌), such that (6)f(z)=g(ω(z))(z𝕌). Indeed, it is known that (7)f(z)g(z)(z𝕌)f(0)=g(0),f(𝕌)g(𝕌). Furthermore, if the function g is univalent in 𝕌, then we have the following equivalence: (8)f(z)g(z)(z𝕌)f(0)=g(0),f(𝕌)g(𝕌).

In a recent paper, Miller et al.  proved the following result.

Theorem A.

Let n, 0λ1, and (9)M0(λ,n)=n+1-λ(n+1-λ)2+λ2+1-λ. If fΣn satisfies the condition (10)|z2f(z)+(1-λ)zf(z)+λ|<M0(λ,n)(z𝕌), then f𝒮n*.

More recently, Catas  improved Theorem A as follows.

Theorem B.

Let n, 0λ<1, and (11)M(λ,n)=max{M0(λ,n),M1(λ,n)}, where M0(λ,n) is given by (9) and (12)M1(λ,n)=2(n+1-λ)(1-λ)(1-λ)(n-1)+(n+1-λ)2(1-λ)+[(n-1)(1-λ)]2. If fΣn satisfies the condition (13)|z2f(z)+(1-λ)zf(z)+λ|<M(λ,n)(z𝕌), then f𝒮n*.

In this paper, we aim at finding the conditions for starlikeness of the expression |z2f(z)+λzf(z)+1-λ| for λ>1.

For some recent investigations of meromorphic functions, see, for example, the works of  and the references cited therein.

In order to prove our main results, we require the following subordination result due to Hallenbeck and Ruscheweyh .

Lemma 1.

Let ϕ be a convex function with ϕ(0)=1, and let γ0 be a complex number with (γ)0. If a function (14)𝔭(z)=1+𝔭nzn+𝔭n+1zn+1+ satisfies the condition (15)𝔭(z)+1γz𝔭(z)ϕ(z), then (16)𝔭(z)χ(z):=γnzγ/n0zϕ(t)t(γ/n)-1dtϕ(z).

2. Main Results

We begin by stating the following result.

Theorem 2.

Let n, λ>1, and 0α<1. If fΣn satisfies the inequality (17)|z2f(z)+λzf(z)+1-λ|<M, where (18)M:=M(λ,α,n)=(1-α)(λ+n-1)λ-α+(1-λ)2+(λ+n-1)2, then f𝒮n*(α).

Proof.

Suppose that (19)q(z):=zf(z)(z𝕌). It follows from (19) that (20)zq(z)=zf(z)+z2f(z). By combining (17), (19), and (20), we easily get (21)|q(z)+1λ-1zq(z)-1|<Mλ-1, or equivalently (22)q(z)+1λ-1zq(z)1+Mλ-1z. An application of Lemma 1 yields (23)q(z)λ-1nz(λ-1)/n0z(1+Mλ-1t)t[(λ-1)/n]  -1dt=1+Mλ+n-1z. The subordination (23) is equivalent to (24)|q(z)-1|<Mλ+n-1=:N. From (18) and (24), we know that (25)N<1-αλ-α<1.

We suppose that (26)-zf(z)f(z):=(1-α)p(z)+α. By virtue of (19) and (26), we get (27)z2f(z)=-q(z)[(1-α)p(z)+α], which implies that (17) can be written as (28)|q(z)[(1-α)p(z)+α-λ]+λ-1|<M=(λ+n-1)N.

We now only need to show that (28) implies (p(z))>0 in 𝕌. Indeed, if this is false, since p(0)=1, then there exists a point z0𝕌 such that p(z0)=βi, where β is a real number. Thus, in order to show that (28) implies (p(z))>0 in 𝕌, it suffices to obtain the contradiction from the inequality (29)|q(z0)[(1-α)βi+α-λ]+λ-1|(λ+n-1)N(β). By setting (30)q(z0)=u+iv(u,v), we have (31)E=|q(z0)[(1-α)βi+α-λ]+λ-1|2=(u2+v2)[(1-α)2β2+(α-λ)2]-2(1-λ)((u+iv)[(1-α)βi+α-λ])+(1-λ)2=(u2+v2)(1-α)2β2+2(1-λ)(1-α)βv+|(u+iv)(α-λ)-(1-λ)|2. By means of (24), we obtain (32)|(u+iv)(α-λ)-(1-λ)|=|(u+iv)(α-λ)-(α-λ)+α-λ-1+λ|=|(α-λ)(u+iv-1)-(1-α)|1-α-(λ-α)|u+iv-1|1-α-(λ-α)N. It follows from (31) and (32) that (33)E(u2+v2)(1-α)2β2+2(1-λ)(1-α)βv+[1-α-(λ-α)N]2.

We now set (34)F(β)E-M2(u2+v2)(1-α)2β2+2(1-λ)(1-α)vβ+[1-α-(λ-α)N]2-(λ+n-1)2N2. If F(β)0, then (29) holds true. Since (u2+v2)(1-α)2>0, the inequality F(β)0 holds if the discriminant Δ0; that is, (35)Δ=(1-α)2×{(1-λ)2v2-(u2+v2)×[(1-α-(λ-α)N)2-(λ+n-1)2N2]}0, and the last inequality is equivalent to (36)v2[(1-λ)2-(1-α-(λ-α)N)2+(λ+n-1)2N2]u2[(1-α-(λ-α)N)2-(λ+n-1)2N2]. Furthermore, in view of (24) and (36), after a geometric argument, we deduce that (37)v2u2N21-N2(1-α-(λ-α)N)2-(λ+n-1)2N2(1-λ)2-(1-α-(λ-α)N)2+(λ+n-1)2N2. It follows from (37) that Δ0, which implies that F(β)0. But this contradicts (28). Therefore, we know that (p(z))>0 in 𝕌. By virtue of (26), we conclude that (38)(zf(z)f(z))<-((1-α)p(z)+α)<-α. This evidently completes the proof of Theorem 2.

Taking α=0 in Theorem 2, we obtain the following result.

Corollary 3.

Let n and λ>1. If fΣn satisfies the inequality (39)|z2f(z)+λzf(z)+1-λ|<(λ+n-1)λ+(1-λ)2+(λ+n-1)2, then f𝒮n*.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The present investigation was supported by the National Natural Science Foundation under Grants nos. 11301008, 11226088, and 11101053, the Foundation for Excellent Youth Teachers of Colleges and Universities of Henan Province under Grant no. 2013GGJS-146, and the Natural Science Foundation of Educational Committee of Henan Province under Grant no. 14B110012 of China.

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