On Certain New Subclass of Meromorphic Close-to-Convex Functions

Abstract

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 𝑓(𝑧)=1𝑧+^∞_𝑛=1𝑎_𝑛𝑧^𝑛, (1) which are analytic in the punctured open unit disk: 𝕌^∗∶={𝑧∶𝑧∈ℂand0<|𝑧|<1}∶=𝕌⧵{0}. (2)

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

A function 𝑓∈Σ is said to be in the class ℳ𝒮^∗(𝛼) of meromorphic starlike functions of order 𝛼 if it satisfies the following inequality: ℜ−𝑧𝑓^′(𝑧)𝑓(𝑧)>𝛼(0≦𝛼<1;𝑧∈𝕌). (5) Moreover, a function 𝑓∈Σ is said to be in the class ℳ𝒞 of meromorphic close-to-convex functions if it satisfies the following condition: ℜ−𝑧𝑓^′(𝑧)𝑔(𝑧)>0𝑧∈𝕌;𝑔∈ℳ𝒮^∗(0)=∶ℳ𝒮^∗. (6)

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)(𝑧∈𝕌), 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 𝑓(𝑧)(𝑧)(0)(0),𝑓(𝕌)(𝕌).≺𝑔)(𝑧∈𝕌⟹𝑓=𝑔⊂𝑔 (7) Furthermore, if the function 𝑔 is univalent in 𝕌, then we have the following equivalence: 𝑓(𝑧)(𝑧)(0)(0),𝑓(𝕌)(𝕌).≺𝑔)(𝑧∈𝕌⟺𝑓=𝑔⊂𝑔 (8)

Recently, Wang et al. [1] introduced and investigated the class ℳ𝒦 of meromorphic close-to-convex functions which satisfy the inequality ℜ𝑓^′(𝑧)𝑔(𝑧)𝑔()(),−𝑧>0𝑧∈𝕌 (9) where 𝑔∈ℳ𝒮^∗(1/2). Kowalczyk and Leś-Bomba [2] discussed the class 𝒦_𝑠(𝛾) of analytic functions related to the starlike functions; a function 𝑓(𝑧)=𝑧+∑^∞_𝑛=2𝑎_𝑛𝑧^𝑛 which is analytic in 𝕌 is said to be in the class 𝒦_𝑠(𝛾)(0≦𝛾<1), if it is satisfies the following inequality: ℜ𝑧²𝑓^′(𝑧)𝑔(𝑧)𝑔()(),−𝑧>𝛾𝑧∈𝕌 (10) where 𝑔∈𝒮^∗(1/2). Şeker [3] discussed the class 𝒦_𝑠^(𝑘)(𝛾) of analytic functions which satisfy the following condition: ℜ𝑧^𝑘𝑓^′(𝑧)𝑔_𝑘(𝑧)>𝛾(𝑧∈𝕌), (11) where 0≦𝛾<1,𝑔_𝑘(𝑧)=∏^𝑘−1_𝑣=0𝜀^−𝑣𝑔(𝜀^𝑣𝑧)(𝜀^𝑘=1;𝑘≧1), and 𝑔∈𝒮^∗((𝑘−1)/𝑘).

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

Definition 1. A function 𝑓∈Σ is said to be in the class ℳ𝒦^(𝑘)(𝛾) if it satisfies the following inequality: ℜ−𝑓^′(𝑧)𝑧^𝑘−2𝑔_𝑘(𝑧)>𝛾(𝑧∈𝕌), (12) where 0≦𝛾<1,𝑘≧1 is a fixed positive integer, and 𝑔_𝑘(𝑧) is given by 𝑔_𝑘(𝑧)=^𝑘−1_𝑣=0𝜀^𝑣𝑔𝜀^𝑣𝑧𝜀=𝑒^{(2𝜋𝑖)/𝑘};𝑔∈ℳ𝒮^∗𝑘−1𝑘. (13)

We observe that the inequality (12) is equivalent to 𝑓^′(𝑧)𝑧^𝑘−2𝑔_𝑘(𝑧)<𝑓+1^′(𝑧)𝑧^𝑘−2𝑔_𝑘(𝑧)+2𝛾−1(𝑧∈𝕌). (14)

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

For some recent investigations on the class of close-to-convex functions, one can find them in [4–7] and the references cited therein. In the present paper, we aim at proving that the class ℳ𝒦^(𝑘)(𝛾) is a subclass of meromorphic close-to-convex functions. Furthermore, some interesting results of the class ℳ𝒦^(𝑘)(𝛾) are derived.

2. Preliminary Results

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

Lemma 2. Let 𝜑_𝑗∈ℳ𝒮^∗(𝛼_𝑗), where 0≦𝛼_𝑗<1(𝑗=0,1,…,𝑘−1). Then for 𝑘−1≦∑^𝑘−1_𝑗=0𝛼_𝑗<𝑘, one has 𝑧^{𝑘−1𝑘−1}_𝑗=0𝜑_𝑗(𝑧)∈ℳ𝒮^∗𝑘−1_𝑗=0𝛼_𝑗−(𝑘−1). (15)

Proof. Since 𝜑_𝑗∈ℳ𝒮^∗(𝛼_𝑗)(𝑗=0,1,…,𝑘−1), we have ℜ−𝑧𝜑^′₀(𝑧)𝜑₀(𝑧)>𝛼₀,ℜ−𝑧𝜑^′₁(𝑧)𝜑₁(𝑧)>𝛼₁,…,ℜ−𝑧𝜑^′_𝑘−1(𝑧)𝜑_𝑘−1(𝑧)>𝛼_𝑘−1. (16) We now let 𝐹(𝑧)=𝑧^𝑘−1𝜑₀(𝑧)𝜑₁(𝑧)⋯𝜑_𝑘−1(𝑧). (17) Differentiating (17) logarithmically, we obtain −𝑧𝐹^′(𝑧)𝐹(𝑧)=−𝑧𝜑^′₀(𝑧)𝜑₀(𝑧)−𝑧𝜑^′₁(𝑧)𝜑₁(𝑧)−⋯−𝑧𝜑^′_𝑘−1(𝑧)𝜑_𝑘−1(𝑧)−(𝑘−1). (18) From (18) together with (16), we can get ℜ−𝑧𝐹^′(𝑧)𝐹(𝑧)=ℜ−𝑧𝜑^′₀(𝑧)𝜑₀(𝑧)+ℜ−𝑧𝜑^′₁(𝑧)𝜑₁(𝑧)+⋯+ℜ−𝑧𝜑^′_𝑘−1(𝑧)𝜑_𝑘−1(𝑧)−()>𝑘−1_{𝑘−1𝑗=0}𝛼_𝑗−(𝑘−1). (19) Thus, if 0≦∑^𝑘−1_𝑗=0𝛼_𝑗−(𝑘−1)<1, we know that 𝐹(𝑧)=𝑧^{𝑘−1𝑘−1}_𝑗=0𝜑_𝑗(𝑧)∈ℳ𝒮^∗𝑘−1_𝑗=0𝛼_𝑗−(𝑘−1). (20)

Lemma 3 (see [8]). Let −1≦𝐵₂≦𝐵₁<𝐴₁≦𝐴₂≦1. Then 1+𝐴₁𝑧1+𝐵₁𝑧≺1+𝐴₂𝑧1+𝐵₂𝑧. (21)

Lemma 4 (see [9]). Suppose that 𝑔∈ℳ𝒮^∗. Then (1−𝑟)²𝑟≦𝑔(𝑧≦())1+𝑟²𝑟(|𝑧|=𝑟;0<𝑟<1). (22)

Lemma 5 (see [10, page 105]). If the function 𝑝(𝑧)=1+^∞_𝑛=1𝑝_𝑛𝑧^𝑛(𝑧∈𝕌) (23) analytic and convex in 𝕌 and satisfies the condition ℜ𝑝(𝑧)>𝛾(𝑧∈𝕌), (24) then 1−1−2𝛾𝑟≦𝑝(𝑧≦𝑟1+𝑟)1+1−2𝛾𝑧|).1−𝑟(|=𝑟<1 (25)

Lemma 6 (see [10]). If the function 𝑝∈𝒫 is given by (3), then 𝑝_𝑛≦2(𝑛∈ℕ),1−𝑟≦𝑝(𝑧≦1+𝑟)1+𝑟𝑧|).1−𝑟(|=𝑟;0<𝑟<1 (26)

Lemma 7 (see [11]). Suppose that 𝑔(𝑧)=1𝑧+^∞_𝑛=1𝑐_𝑛𝑧^𝑛∈ℳ𝒮^∗. (27) Then 𝑐_𝑛≦2({.𝑛+1𝑛∈ℕ∶=1,2,…}) (28) Each of these inequalities is sharp, with the extremal function given by 𝑔(𝑧)=𝑧⁻¹1+𝑧^{𝑛+12/(𝑛+1)}. (29)

Lemma 8 (see [12]). Let 𝑓(𝑧)=1+^∞_𝑛=1𝑐_𝑛𝑧^𝑛 (30) be analytic in 𝕌 and let 𝑔(𝑧)=1+^∞_𝑛=1𝑑_𝑛𝑧^𝑛 (31) be analytic and convex in 𝕌. If 𝑓≺𝑔, then 𝑐_𝑛≦𝑑₁(𝑛∈ℕ∶={1,2,…}). (32)

Lemma 9. If 𝑝(𝑧)≺(1+(1−2𝛾)𝑧)/(1−𝑧), where 𝑝(𝑧) is given by (3), then 𝑝_𝑛≦21−𝛾. (33)

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=1𝑏_𝑛𝑧^𝑛∈ℳ𝒮^∗((𝑘−1)/𝑘), Then 𝐺_𝑘(𝑧)=𝑧^𝑘−1𝑔_𝑘(𝑧)=1𝑧+^∞_𝑛=1𝐵_𝑛𝑧^𝑛∈ℳ𝒮^∗, (34) where 𝑔_𝑘(𝑧) is given by (13).

Proof. From (13), we know that 𝑧^𝑘−1𝑔_𝑘(𝑧)=𝑧^{𝑘−1𝑘−1}_𝑣=0𝜀^𝑣𝑔𝜀^𝑣𝑧=𝑧^{𝑘−1𝑘−1}_𝑣=0𝜀^𝑣1𝜀^𝑣𝑧+^∞_𝑛=1𝑏_𝑛𝜀^𝑣𝑧^𝑛=𝑧^{𝑘−1𝑘−1}_𝑣=01𝑧+^∞_𝑛=1𝑏_𝑛𝜀^(𝑛+1)𝑣𝑧^𝑛. (35) Now, suppose that 𝑔(𝑧)=1𝑧+^∞_𝑛=1𝑏_𝑛𝑧^𝑛∈ℳ𝒮^∗𝑘−1𝑘. (36) 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 𝑓∈ℳ𝒦^𝑘(𝛾), then 𝑓 is a meromorphic close-to-convex function. So the class ℳ𝒦^𝑘(𝛾) is a subclass of meromorphic close-to-convex functions.

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

Theorem 12. Let 𝑓∈Σ,𝑔∈ℳ𝒮^∗((𝑘−1)/𝑘). If for 0≦𝛾<1, one has 2^∞_𝑛=1𝑛𝑎_𝑛+1−2𝛾+1^∞_𝑛=1𝐵_𝑛≦21−𝛾, (37) where the coefficients 𝐵_𝑛 (𝑛=1,2,…) are given by (34), then 𝑓∈ℳ𝒦^𝑘(𝛾).

Proof. We set for 𝑓 given by (1) and 𝑔_𝑘 defined by (13) Δ=−𝑧𝑓^′(𝑧)−𝑧^𝑘−1𝑔_𝑘(𝑧−)−𝑧𝑓^′(𝑧)+1−2𝛾𝑧^𝑘−1𝑔_𝑘(𝑧=−_∞𝑛=1)𝑛𝑎_𝑛𝑧^𝑛−^∞_𝑛=1𝐵_𝑛𝑧^𝑛−2−2𝛾𝑧−^∞_𝑛=1𝑛𝑎_𝑛𝑧^𝑛+1−2𝛾^∞_𝑛=1𝐵_𝑛𝑧^𝑛. (38) For |𝑧|=𝑟(0<𝑟<1), from inequality (37), we have Δ≦^∞_𝑛=1𝑛𝑎_𝑛|𝑧|^𝑛+^∞_𝑛=1𝐵_𝑛|𝑧|^𝑛−12−2𝛾|𝑧|−^∞_𝑛=1𝑛𝑎_𝑛|𝑧|^𝑛−1−2𝛾^∞_𝑛=1𝐵_𝑛|𝑧|^𝑛1=−2−2𝛾|𝑧|+2^∞_𝑛=1𝑛𝑎_𝑛|𝑧|^𝑛+×1−2𝛾+1_∞𝑛=1𝐵_𝑛|𝑧|^𝑛<−2−2𝛾+2^∞_𝑛=1𝑛𝑎_𝑛+1−2𝛾+1^∞_𝑛=1𝐵_𝑛1|𝑧|≦0. (39) Thus, we have −𝑓^′(𝑧)𝑧^𝑘−2𝑔_𝑘(𝑧)<−𝑓−1^′(𝑧)𝑧^𝑘−2𝑔_𝑘(𝑧)+1−2𝛾(𝑧∈𝕌), (40) that is, 𝑓∈ℳ𝒦^(𝑘)(𝛾). This completes the proof of Theorem 12.

Next, we give the inclusion relationship for class 𝑓∈ℳ𝒦^(𝑘)(𝛾).

Theorem 13. Let 0≦𝛾₁≦𝛾₂<1. Then one has ℳ𝒦^(𝑘)𝛾₂⊂ℳ𝒦^(𝑘)𝛾₁. (41)

Proof. Suppose that 𝑓∈ℳ𝒦^(𝑘)(𝛾₂). By Definition 1, we have −𝑓^′(𝑧)𝑧^𝑘−2𝑔_𝑘(𝑧)≺1+1−2𝛾₂𝑧.1−𝑧 (42) Since 0≦𝛾₁≦𝛾₂<1, we get −1<1−2𝛾₂≦1−2𝛾₁≦1. (43) Thus, by Lemma 3, we obtain −𝑓^′(𝑧)𝑧^𝑘−2𝑔_𝑘(𝑧)≺1+1−2𝛾₁𝑧,1−𝑧 (44) that is, 𝑓∈ℳ𝒦^(𝑘)(𝛾₁). This means that ℳ𝒦^(𝑘)(𝛾₂)⊂ℳ𝒦^(𝑘)(𝛾₁). Hence the proof is completed.

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

Theorem 14. Suppose that 𝑓(𝑧)=1𝑧+^∞_𝑛=1𝑎_𝑛𝑧^𝑛∈ℳ𝒦^(𝑘)𝛾. (45) Then 𝑎₁𝑎≦1,_𝑛≦2𝑛()+𝑛+12−2𝛾𝑛1+^𝑛−1_𝑗=12{1.𝑗+1(𝑛∈ℕ⧵}) (46)(47)

Proof. Suppose that 𝑓∈ℳ𝒦^(𝑘)(𝛾); we know that ℜ−𝑧𝑓^′(𝑧)𝐺_𝑘(𝑧)>𝛾, (48) where 𝐺_𝑘 is given by (34). If we set 𝑞(𝑧)∶=−𝑧𝑓^′(𝑧)𝐺_𝑘(𝑧), (49) it follows that 𝑞(𝑧)=1+^∞_𝑛=1𝑑_𝑛𝑧^𝑛. (50) In view of Definition 1 and Lemma 9, we know that 𝑑_𝑛≦2−2𝛾(𝑛∈ℕ). (51) By substituting the series expressions of functions 𝑓,𝐺_𝑘, and 𝑞 into (49), we obtain 1+^∞_𝑛=1𝑑_𝑛𝑧^𝑛1𝑧+^∞_𝑛=1𝐵_𝑛𝑧^𝑛=1𝑧−^∞_𝑛=1𝑛𝑎_𝑛𝑧^𝑛. (52) Since 𝑓 is univalent in 𝕌^∗, it is well known that 𝑎₁≦1. (53) On the other hand, we find from (52) that −𝑛𝑎_𝑛=𝐵_𝑛+𝑑_𝑛+1+^𝑛−1_𝑗=1𝐵_𝑗𝑑_𝑛−𝑗(𝑛∈ℕ). (54) Combining (28), (51), and (54), we have 𝑛𝑎_𝑛≦2++𝑛+12−2𝛾2−2𝛾^𝑛−1_𝑗=12.𝑗+1 (55) Thus, the assertion (47) of Theorem 14 follows directly from (55).

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

Theorem 15. If 𝑓∈ℳ𝒦^(𝑘)(𝛾), then (1−𝑟)²1−1−2𝛾𝑟𝑟²()≦𝑓1+𝑟^′(𝑧≦())1+𝑟²1+1−2𝛾𝑟𝑟²()𝑧|),1−𝑟(|=𝑟<1_𝑟0(1−𝑡)²1−1−2𝛾𝑡𝑡²()𝑓(𝑧≦1+𝑡𝑑𝑡≦)_𝑟0(1+𝑡)²1+1−2𝛾𝑡𝑡²(1−𝑡)𝑑𝑡. (56)(57)

Proof. If 𝑓∈ℳ𝒦^(𝑘)(𝛾), then there exists a function 𝑔∈ℳ𝒮^∗((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 (1−𝑟)²𝑟≦𝐺_𝑘(𝑧≦())1+𝑟²𝑟(|𝑧|=𝑟;0<𝑟<1). (58) Let 𝑞 be defined by (49); by Lemma 5, we know that 1−1−2𝛾𝑟≦𝑞(𝑧≦𝑟1+𝑟)1+1−2𝛾𝑧|).1−𝑟(|=𝑟<1 (59) 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.