JCA Journal of Complex Analysis 2314-4971 2314-4963 Hindawi Publishing Corporation 704784 10.1155/2013/704784 704784 Research Article On Certain New Subclass of Meromorphic Close-to-Convex Functions Yi Jing-Ping 1 Wang Zhi-Gang 1 Zeng Ming-Hua 2 Lindstrom Mikael 1 School of Mathematics and Statistics Anyang Normal University Anyang, Henan 455002 China aynu.edu.cn 2 School of Railway Tracks and Transportation East China Jiao Tong University Nanchang, Jiangxi 330013 China ecjtu.jx.cn 2013 11 3 2013 2013 19 11 2012 31 01 2013 2013 Copyright © 2013 Jing-Ping Yi et al. 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.

We introduce a certain new subclass of meromorphic close-to-convex functions. Such results as inclusion relationship, coefficient inequalities, distortion, and growth theorems for this class of functions are derived.

1. Introduction

Let Σ denote the class of functions of the form (1)f(z)=1z+n=1anzn, which are analytic in the punctured open unit disk: (2)𝕌*={z:zand0<|z|<1}=𝕌{0}.

Let 𝒫 denote the class of functions p given by (3)p(z)=1+n=1pnzn(z𝕌), which are analytic and convex in 𝕌 and satisfy the following condition: (4)(p(z))>0(z𝕌).

A function fΣ is said to be in the class 𝒮*(α) of meromorphic starlike functions of order α if it satisfies the following inequality: (5)(-zf(z)f(z))>α(0α<1;z𝕌). Moreover, a function fΣ is said to be in the class 𝒞 of meromorphic close-to-convex functions if it satisfies the following condition: (6)(-zf(z)g(z))>0(z𝕌;g𝒮*(0)=:𝒮*).

For two functions f and g analytic in 𝕌, we say that the function f(z) is subordinate to g(z) in 𝕌 and write f(z)g(z)(z𝕌), if there exists a Schwarz function ω(z), analytic in 𝕌 with ω(0)=0 and |ω(z)|<1 such that f(z)=g(ω(z)). Indeed, it is well 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).

Recently, Wang et al.  introduced and investigated the class 𝒦 of meromorphic close-to-convex functions which satisfy the inequality (9)(f(z)g(z)g(-z))>0(z𝕌), where g𝒮*(1/2). Kowalczyk and Leś-Bomba  discussed the class 𝒦s(γ) of analytic functions related to the starlike functions; a function f(z)=z+n=2anzn which is analytic in 𝕌 is said to be in the class 𝒦s(γ)(0γ<1), if it is satisfies the following inequality: (10)(z2f(z)g(z)g(-z))>γ(z𝕌), where g𝒮*(1/2). Şeker  discussed the class 𝒦s(k)(γ) of analytic functions which satisfy the following condition: (11)(zkf(z)gk(z))>γ(z𝕌), where 0γ<1,gk(z)=v=0k-1ε-vg(εvz)(εk=1;k1), and g𝒮*((k-1)/k).

Motivated essentially by the classes 𝒦,𝒦s(γ), and 𝒦s(k)(γ), we introduce and study the following more generalized class 𝒦(k)(γ) of meromorphic functions.

Definition 1.

A function fΣ is said to be in the class 𝒦(k)(γ) if it satisfies the following inequality: (12)(-f(z)zk-2gk(z))>γ(z𝕌), where 0γ<1,k1 is a fixed positive integer, and gk(z) is given by (13)gk(z)=v=0k-1εvg(εvz)(ε=e(2πi)/k;g𝒮*(k-1k)).

We observe that the inequality (12) is equivalent to (14)|f(z)zk-2gk(z)+1|<|f(z)zk-2gk(z)+2γ-1|(z𝕌).

Since 𝒦(2)(0)=𝒦, the class 𝒦(k)(γ) is a generalization of the class 𝒦.

For some recent investigations on the class of close-to-convex functions, one can find them in  and the references cited therein. In the present paper, we aim at proving that the class 𝒦(k)(γ) is a subclass of meromorphic close-to-convex functions. Furthermore, some interesting results of the class 𝒦(k)(γ) are derived.

2. Preliminary Results

To prove our main results, we need the following lemmas.

Lemma 2.

Let φj𝒮*(αj), where 0αj<1(j=0,1,,k-1). Then for k-1j=0k-1αj<k, one has (15)zk-1j=0k-1φj(z)𝒮*(j=0k-1αj-(k-1)).

Proof.

Since φj𝒮*(αj)(j=0,1,,k-1), we have (16)(-zφ0(z)φ0(z))>α0,(-zφ1(z)φ1(z))>α1,,(-zφk-1(z)φk-1(z))>αk-1. We now let (17)F(z)=zk-1φ0(z)φ1(z)φk-1(z). Differentiating (17) logarithmically, we obtain (18)-zF(z)F(z)=-zφ0(z)φ0(z)-zφ1(z)φ1(z)--zφk-1(z)φk-1(z)-(k-1). From (18) together with (16), we can get (19)(-zF(z)F(z))=(-zφ0(z)φ0(z))+(-zφ1(z)φ1(z))++(-zφk-1(z)φk-1(z))-(k-1)>j=0k-1αj-(k-1). Thus, if 0j=0k-1αj-(k-1)<1, we know that (20)F(z)=zk-1j=0k-1φj(z)𝒮*(j=0k-1αj-(k-1)).

Lemma 3 (see [<xref ref-type="bibr" rid="B5">8</xref>]).

Let -1B2B1<A1A21. Then (21)1+A1z1+B1z1+A2z1+B2z.

Lemma 4 (see [<xref ref-type="bibr" rid="B7">9</xref>]).

Suppose that g𝒮*. Then (22)(1-r)2r|g(z)|(1+r)2r(|z|=r;  0<r<1).

Lemma 5 (see [<xref ref-type="bibr" rid="B2">10</xref>, page 105]).

If the function (23)p(z)=1+n=1pnzn(z𝕌) analytic and convex in 𝕌 and satisfies the condition (24)(p(z))>γ(z𝕌), then (25)1-(1-2γ)r1+r|p(z)|1+(1-2γ)r1-r(|z|=r<1).

Lemma 6 (see [<xref ref-type="bibr" rid="B2">10</xref>]).

If the function p𝒫 is given by (3), then (26)|pn|2(n),1-r1+r|p(z)|1+r1-r(|z|=r;  0<r<1).

Lemma 7 (see [<xref ref-type="bibr" rid="B1">11</xref>]).

Suppose that (27)g(z)=1z+n=1cnzn𝒮*. Then (28)|cn|2n+1(n={1,2,}). Each of these inequalities is sharp, with the extremal function given by (29)g(z)=z-1(1+zn+1)2/(n+1).

Lemma 8 (see [<xref ref-type="bibr" rid="B8">12</xref>]).

Let (30)f(z)=1+n=1cnzn be analytic in 𝕌 and let (31)g(z)=1+n=1dnzn be analytic and convex in 𝕌. If fg, then (32)|cn||d1|(n={1,2,}).

Lemma 9.

If p(z)(1+(1-2γ)z)/(1-z), where p(z) is given by (3), then (33)|pn|2(1-γ).

Proof.

By Lemma 8, we easily get the assertion of Lemma 9.

3. Main Results

We first give the following result.

Theorem 10.

Let g(z)=1/z+n=1bnzn𝒮*((k-1)/k), Then (34)Gk(z)=zk-1gk(z)=1z+n=1Bnzn𝒮*, where gk(z) is given by (13).

Proof.

From (13), we know that (35)zk-1gk(z)=zk-1v=0k-1εvg(εvz)=zk-1v=0k-1εv(1εvz+n=1bn(εvz)n)=zk-1v=0k-1(1z+n=1bnε(n+1)vzn). Now, suppose that (36)g(z)=1z+n=1bnzn𝒮*(k-1k). Then, by Lemma 2 and (35), we get the assertion of Theorem 10 easily.

Remark 11.

From Theorem 10 and Definition 1, we know that if f𝒦k(γ), then f is a meromorphic close-to-convex function. So the class 𝒦k(γ) is a subclass of meromorphic close-to-convex functions.

Now, we prove a sufficient condition for functions to belong to the class 𝒦k(γ).

Theorem 12.

Let fΣ,g𝒮*((k-1)/k). If for 0γ<1, one has (37)2n=1n|an|+(|1-2γ|+1)n=1|Bn|2(1-γ), where the coefficients Bn (n=1,2,) are given by (34), then f𝒦k(γ).

Proof.

We set for f given by (1) and gk defined by (13) (38)Δ=|-zf(z)-zk-1gk(z)|-|-zf(z)+(1-2γ)zk-1gk(z)|=|-n=1nanzn-n=1Bnzn|-|2-2γz-n=1nanzn+(1-2γ)n=1Bnzn|. For |z|=r  (0<r<1), from inequality (37), we have (39)Δn=1n|an||z|n+n=1|Bn||z|n-((2-2γ)1|z|-n=1n|an||z|n-|1-2γ|n=1|Bn||z|n)=-(2-2γ)1|z|+2n=1n|an||z|n+(|1-2γ|+1)×n=1|Bn||z|n<(-(2-2γ)+2n=1n|an|+(|1-2γ|+1)n=1|Bn|)1|z|0. Thus, we have (40)|-f(z)zk-2gk(z)-1|<|-f(z)zk-2gk(z)+1-2γ|(z𝕌), that is, f𝒦(k)(γ). This completes the proof of Theorem 12.

Next, we give the inclusion relationship for class f𝒦(k)(γ).

Theorem 13.

Let 0γ1γ2<1. Then one has (41)𝒦(k)(γ2)𝒦(k)(γ1).

Proof.

Suppose that f𝒦(k)(γ2). By Definition 1, we have (42)-f(z)zk-2gk(z)1+(1-2γ2)z1-z. Since 0γ1γ2<1, we get (43)-1<1-2γ21-2γ11. Thus, by Lemma 3, we obtain (44)-f(z)zk-2gk(z)1+(1-2γ1)z1-z, that is, f𝒦(k)(γ1). This means that 𝒦(k)(γ2)𝒦(k)(γ1). Hence the proof is completed.

In what follows, we derive the coefficient inequality for the class 𝒦(k)(γ).

Theorem 14.

Suppose that (45)f(z)=1z+n=1anzn𝒦(k)(γ). Then (46)|a1|1,(47)|an|2n(n+1)+(2-2γ)n(1+j=1n-12j+1)(n{1}).

Proof.

Suppose that f𝒦(k)(γ); we know that (48)(-zf(z)Gk(z))>γ, where Gk is given by (34). If we set (49)q(z)=-zf(z)Gk(z), it follows that (50)q(z)=1+n=1dnzn. In view of Definition 1 and Lemma 9, we know that (51)|dn|(2-2γ)(n). By substituting the series expressions of functions f,Gk, and q into (49), we obtain (52)(1+n=1dnzn)(1z+n=1Bnzn)=(1z-n=1nanzn). Since f is univalent in 𝕌*, it is well known that (53)|a1|1. On the other hand, we find from (52) that (54)-nan=Bn+dn+1+j=1n-1Bjdn-j(n). Combining (28), (51), and (54), we have (55)n|an|2n+1+(2-2γ)+(2-2γ)j=1n-12j+1. Thus, the assertion (47) of Theorem 14 follows directly from (55).

Finally, we give the distortion and growth theorems for the function class 𝒦(k)(γ).

Theorem 15.

If f𝒦(k)(γ), then (56)(1-r)2[1-(1-2γ)r]r2(1+r)|f(z)|(1+r)2[1+(1-2γ)r]r2(1-r)(|z|=r<1),(57)0r(1-t)2[1-(1-2γ)t]t2(1+t)dt|f(z)|0r(1+t)2[1+(1-2γ)t]t2(1-t)dt.

Proof.

If f𝒦(k)(γ), then there exists a function g𝒮*((k-1)/k) such that (12) holds. It follows from Theorem 10 that the function given by (34) is a meromorphic starlike function. Therefore, by Lemma 4, we have (58)(1-r)2r|Gk(z)|(1+r)2r(|z|=r;0<r<1). Let q be defined by (49); by Lemma 5, we know that (59)1-(1-2γ)r1+r|q(z)|1+(1-2γ)r1-r(|z|=r<1). Thus, from (49), (58), and (59), we readily get (56). Upon integrating (56) from 0 to r, we get (57). The proof of Theorem 15 is thus completed.

Acknowledgments

The present investigation was supported by the National Natural Science Foundation under Grant 11226088, the Open Fund Project of Key Research Institute of Philosophies and Social Sciences in Hunan Universities under Grants 11FEFM02 and 12FEFM02, the Key Project of Natural Science Foundation of Educational Committee of Henan Province under Grant 12A110002, and the Science and Technology Program of Educational Department of Jiangxi Province under Grant GJJ12322 of China. The authors are grateful to the referees for their valuable comments and suggestions which essentially improved the quality of the paper.

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