A New Class of Meromorphic Functions Associated with Spirallike Functions

Abstract

We introduce a new class of meromorphic functions associated with spirallike functions. Such results as subordination property, integral representation, convolution property, and coefficient inequalities are proved.

1. Introduction

Let Σ denote the class of functions 𝑓 of the form 𝑓(𝑧)=1𝑧+^∞_𝑘=0𝑎_𝑘𝑧^𝑘, (1.1) which are analytic in the punctured open unit disk 𝕌^∗∶={𝑧∶𝑧∈ℂ,0<|𝑧|<1}=∶𝕌⧵{0}. (1.2)

Let 𝒫 denote the class of functions 𝑝 given by 𝑝(𝑧)=1+^∞_𝑘=1𝑝_𝑘𝑧^𝑘(𝑧∈𝕌), (1.3) which are analytic in 𝕌 and satisfy the condition Re𝑝(𝑧)>0(𝑧∈𝕌). (1.4)

Let 𝑓,𝑔∈Σ, where 𝑓 is given by (1.1) and 𝑔 is defined by 𝑔(𝑧)=1𝑧+^∞_𝑘=0𝑏_𝑘𝑧^𝑘, (1.5) then the Hadamard product (or convolution) 𝑓∗𝑔 is defined by (𝑧)1𝑓∗𝑔∶=𝑧+^∞_𝑘=0𝑎_𝑘𝑏_𝑘𝑧^𝑘=∶𝑔∗𝑓(𝑧). (1.6)

For two functions 𝑓 and 𝑔, analytic in 𝕌, we say that the function 𝑓 is subordinate to 𝑔 in 𝕌 and write 𝑓(𝑧)≺𝑔(𝑧)(𝑧∈𝕌), (1.7) if there exists a Schwarz function 𝜔, which is analytic in 𝕌 with 𝜔(0)=0,|𝜔(𝑧)|<1(𝑧∈𝕌), (1.8) such that 𝑓(𝑧)=𝑔(𝜔(𝑧))(𝑧∈𝕌). (1.9) Indeed, it is known that 𝑓(𝑧)≺𝑔(𝑧)(𝑧∈𝕌)⇒𝑓(0)=𝑔(0),𝑓(𝕌)⊂𝑔(𝕌). (1.10) Furthermore, if the function 𝑔 is univalent in 𝕌, then we have the following equivalence: 𝑓(𝑧)≺𝑔(𝑧)(𝑧∈𝕌)⟺𝑓(0)=𝑔(0),𝑓(𝕌)⊂𝑔(𝕌). (1.11)

A function 𝑓∈Σ is said to be in the class ℳ𝒮^∗(𝛽) of meromorphic starlike functions of order 𝛽 if it satisfies the inequality Re𝑧𝑓^′(𝑧)𝑓(𝑧)<−𝛽𝑧∈𝕌;0≦𝛽<1. (1.12)

For the real number 𝛽(0<𝛽<1), we know that 𝑓(𝑧)𝑧𝑓^′(𝑧)+1<12𝛽2𝛽⟺Re𝑧𝑓^′(𝑧)𝑓(𝑧)<−𝛽. (1.13) If the complex number 𝛼 satisfies the condition 1𝛼−2<12, (1.14) it can be easily verified that 𝑓(𝑧)𝑧𝑓^′(𝑧)+1<12𝛼2|𝛼|−1⟺Re𝛼𝑧𝑓^′(𝑧)𝑓(𝑧)>1. (1.15)

We now introduce and investigate the following class of meromorphic functions.

Definition 1.1. A function 𝑓∈Σ is said to be in the class ℳ𝒮_𝛼 if it satisfies the inequality −1Re𝛼𝑧𝑓^′(𝑧)𝑓(𝑧)1>1𝑧∈𝕌;𝛼−2<12. (1.16)

Remark 1.2. For 0<𝛼<1, the class ℳ𝒮_𝛼 is the familiar class of meromorphic starlike functions of order 𝛼.

Remark 1.3. If 𝛼=|𝛼|𝑒^𝑖𝜓(−𝜋/2<𝜓<𝜋/2), then the condition (1.16) is equivalent to Re𝑒^−𝑖𝜓𝑧𝑓^′(𝑧)𝑓(𝑧)<−|𝛼|(𝑧∈𝕌), (1.17) which implies that 𝑓 belongs to the class of meromorphic spirallike functions. Thus, the class of meromorphic spirallike functions is a special case of the class ℳ𝒮_𝛼.
For some recent investigations on spirallike functions and related functions, see, for example, the earlier works [1–9] and the references cited in each of these earlier investigations.

Remark 1.4. The function 𝑓(𝑧)=𝑧⁻¹(1−𝑧)^{2𝛼[Re(1/𝛼)−1]}𝑧∈𝕌^∗;1𝛼−2<12 (1.18) belongs to the class ℳ𝒮_𝛼.
It is clear that 1Re𝛼1>1𝛼−2<12. (1.19) Then, for the function 𝑓 given by (1.18), we know that −1Re𝛼𝑧𝑓^′(𝑧)𝑓(𝑧)1=Re𝛼1+2Re𝛼𝑧−111−𝑧>Re𝛼1−Re𝛼+1=1, (1.20) which implies that 𝑓∈ℳ𝒮_𝛼.
In this paper, we aim at deriving the subordination property, integral representation, convolution property, and coefficient inequalities of the function class ℳ𝒮_𝛼.

2. Preliminary Results

In order to derive our main results, we need the following lemmas.

Lemma 2.1. Let 𝜆 be a complex number. Suppose also that the sequence {𝐴_𝑘}^∞_𝑘=0 is defined by 𝐴₀=2𝜆,𝐴_𝑘+1=2𝜆𝑘+21+^𝑘_𝑙=0𝐴_𝑙𝑘∈ℕ₀∶=ℕ∪{0}. (2.1) Then 𝐴_𝑘=1(𝑘+1)!^𝑘_𝑗=02𝜆+𝑗𝑘∈ℕ₀. (2.2)

Proof. From (2.1), we know that (𝑘+2)𝐴_𝑘+1=2𝜆1+^𝑘_𝑙=0𝐴_𝑙,(𝑘+1)𝐴_𝑘=2𝜆1+^𝑘−1_𝑙=0𝐴_𝑙. (2.3) By virtue of (2.3), we find that 𝐴_𝑘+1𝐴_𝑘=𝑘+1+2𝜆𝑘+2𝑘∈ℕ₀. (2.4) Thus, for 𝑘≧1, we deduce from (2.4) that 𝐴_𝑘=𝐴_𝑘𝐴_𝑘−1⋯𝐴₃𝐴₂⋅𝐴₂𝐴₁⋅𝐴₁𝐴₀⋅𝐴₀=1(𝑘+1)!^𝑘_𝑗=02𝜆+𝑗. (2.5) By virtue of (2.1) and (2.5), we get the desired assertion (2.2) of Lemma 2.1.

Lemma 2.2 (Jack's Lemma [10]). Let 𝜙 be a nonconstant regular function in 𝕌. If |𝜙| attains its maximum value on the circle |𝑧|=𝑟<1 at 𝑧₀, then 𝑧₀𝜙^′𝑧₀=𝑡𝜙𝑧₀, (2.6) for some real number 𝑡(𝑡≧1).

3. Main Results

We begin by deriving the following subordination property of functions belonging to the class ℳ𝒮_𝛼.

Theorem 3.1. A function 𝑓∈ℳ𝒮_𝛼 if and only if −𝑧𝑓^′(𝑧)𝑓(𝑧)1≺1+2𝛼Re𝛼𝑧−11−𝑧𝑧∈𝕌^∗;1𝛼−2<12. (3.1)

Proof. Suppose that ℎ(𝑧)−()∶=1/𝛼𝑧𝑓^′(𝑧)/𝑓(𝑧)−1−𝑖Im(1/𝛼)Re(1/𝛼)−1𝑧∈𝕌;𝑓∈ℳ𝒮_𝛼. (3.2) We easily know that ℎ∈𝒫, which implies that −(1/𝛼)𝑧𝑓^′(𝑧)/𝑓(𝑧)−1−𝑖Im(1/𝛼)()=(𝑧)Re1/𝛼−11+𝜔1−𝜔(𝑧)𝑧∈𝕌;𝑓∈ℳ𝒮_𝛼, (3.3) where 𝜔 is analytic in 𝕌 with 𝜔(0)=0 and |𝜔(𝑧)|<1(𝑧∈𝕌).
It follows from (3.3) that −𝑧𝑓^′(𝑧)𝑓(𝑧)1=1+2𝛼Re𝛼𝜔(𝑧)−1(𝑧)(),1−𝜔𝑧∈𝕌 (3.4) which is equivalent to the subordination relationship (3.1).
On the other hand, the above deductive process can be converse. The proof of Theorem 3.1 is thus completed.

Theorem 3.2. Let 𝑓∈ℳ𝒮_𝛼. Then 𝑓(𝑧)=1𝑧1⋅exp−2𝛼Re𝛼−1^𝑧₀𝜔(𝑡)𝑡(1−𝜔(𝑡))𝑑𝑡𝑧∈𝕌^∗, (3.5) where 𝜔 is analytic in 𝕌 with 𝜔(0)=0 and |𝜔(𝑧)|<1(𝑧∈𝕌).

Proof. For 𝑓∈ℳ𝒮_𝛼, by Theorem 3.1, we know that (3.1) holds true. It follows that −𝑧𝑓^′(𝑧)𝑓(𝑧)1=1+2𝛼Re𝛼𝜔(𝑧)−1(𝑧)(),1−𝜔𝑧∈𝕌 (3.6) where 𝜔 is analytic in 𝕌 with 𝜔(0)=0 and |𝜔(𝑧)|<1(𝑧∈𝕌).
We now find from (3.6) that 𝑓^′(𝑧)𝑓(𝑧)+1𝑧1=−2𝛼Re𝛼𝜔(𝑧)−1𝑧(1−𝜔(𝑧))𝑧∈𝕌^∗, (3.7) which, upon integration, yields (𝑧)1log𝑧𝑓=−2𝛼Re𝛼−1^𝑧₀𝜔(𝑡)𝑡((𝑡().1−𝜔))𝑑𝑡𝑧∈𝕌 (3.8) The assertion (3.5) of Theorem 3.2 can be easily derived from (3.8).

Theorem 3.3. Let 𝑓∈ℳ𝒮_𝛼. Then 𝑓(𝑧)∗1−𝑒^𝑖𝜃𝑧+2𝛼[Re(1/𝛼)−1]𝑒^𝑖𝜃(1−𝑧)𝑧(1−𝑧)²≠0𝑧∈𝕌^∗;0<𝜃<2𝜋. (3.9)

Proof. Assume that 𝑓∈ℳ𝒮_𝛼. By Theorem 3.1, we know that (3.1) holds, which implies that −𝑧𝑓^′(𝑧)𝑓(𝑧)1≠1+2𝛼Re𝛼𝑒−1^𝑖𝜃1−𝑒^𝑖𝜃𝑧∈𝕌^∗;0<𝜃<2𝜋. (3.10) It is easy to see that the condition (3.10) can be written as follows: 1−𝑒^𝑖𝜃𝑧𝑓^′(𝑧)+1−𝑒^𝑖𝜃1+2𝛼Re𝛼−1𝑒^𝑖𝜃𝑓(𝑧)≠0. (3.11) We note that 𝑓(𝑧)(𝑧)∗1=𝑓𝑧𝑧+1+(𝑧)∗11−𝑧=𝑓𝑧(),1−𝑧−𝑧𝑓^′(𝑧)(𝑧)∗1=𝑓𝑧−𝑧(1−𝑧)²=𝑓(𝑧)∗1−2𝑧𝑧(1−𝑧)². (3.12) Thus, by substituting (3.12) into (3.11), we get the desired assertion (3.9) of Theorem 3.3.

Theorem 3.4. Let 𝜆=[Re(1/𝛼)−1]|𝛼|. If 𝑓∈ℳ𝒮_𝛼, then 𝑎_𝑘≦1(𝑘+1)!^𝑘_𝑗=02𝜆+𝑗𝑘∈ℕ₀. (3.13) The inequality (3.13) is sharp for the function given by 𝑓(𝑧)=1𝑧(1−𝑧)^2−2𝛼(0<𝛼<1). (3.14)

Proof. Suppose that ℎ(𝑧)−()∶=1/𝛼𝑧𝑓^′(𝑧)/𝑓(𝑧)−1−𝑖Im(1/𝛼)().Re1/𝛼−1 (3.15) We easily know that ℎ∈𝒫.
If we put ℎ(𝑧)=1+ℎ₁𝑧+ℎ₂𝑧²+⋯, (3.16) it is known that ℎ_𝑘≦2(𝑘∈ℕ). (3.17) From (3.15), we have −1𝛼𝑧𝑓^′(𝑧)𝑓(𝑧)1−1−𝑖Im𝛼=1Re𝛼−1ℎ(𝑧). (3.18) We now set 1𝐴∶=1+𝑖Im𝛼,1𝐵∶=Re𝛼−1. (3.19) It follows from (3.18) that −𝑧𝑓^′(𝑧)=[𝛼𝐴+𝛼𝐵ℎ(𝑧)]𝑓(𝑧). (3.20) Combining (1.1), (3.16), and (3.20), we obtain −1−𝑧𝑧²+𝑎₁+2𝑎₂𝑧+⋯+𝑘𝑎_𝑘𝑧^𝑘−1=+⋯1+𝛼𝐵ℎ₁𝑧+⋯+𝛼𝐵ℎ_𝑘𝑧^𝑘1+⋯𝑧+𝑎₀+𝑎₁𝑧+⋯+𝑎_𝑘𝑧^𝑘+⋯. (3.21) In view of (3.21), we get 𝑎₀+𝛼𝐵ℎ₁=0,−𝑘𝑎_𝑘=𝑎_𝑘+𝛼𝐵𝑎_𝑘−1ℎ₁+𝑎_𝑘−2ℎ₂+⋯+𝑎₀ℎ_𝑘+ℎ_𝑘+1(𝑘∈ℕ). (3.22)(3.23) From (3.17) and (3.22), we obtain 𝑎₀≦2|𝛼|𝐵=2𝜆. (3.24) Moreover, we deduce from (3.17) and (3.23) that 𝑎_𝑘≦2|𝛼|𝐵𝑘+11+^𝑘−1_𝑙=0𝑎_𝑙=2𝜆𝑘+11+^𝑘−1_𝑙=0𝑎_𝑙(𝑘∈ℕ). (3.25)
Next, we define the sequence {𝐴_𝑘}^∞_𝑘=0 as follows: 𝐴₀=2𝜆,𝐴_𝑘+1=2𝜆𝑘+21+^𝑘_𝑙=0𝐴_𝑙𝑘∈ℕ₀. (3.26)
In order to prove that 𝑎_𝑘≦𝐴_𝑘, (3.27) we make use of the principle of mathematical induction. By noting that 𝑎₀≦2𝜆=𝐴₀. (3.28) Therefore, assuming that 𝑎_𝑙≦𝐴_𝑙𝑙=0,1,2,…,𝑘;𝑘∈ℕ₀. (3.29) Combining (3.25) and (3.26), we get 𝑎_𝑘+1≦2𝜆𝑘+21+^𝑘_𝑙=0𝑎_𝑙≦2𝜆𝑘+21+^𝑘_𝑙=0𝐴_𝑙=𝐴_𝑘+1. (3.30) Hence, by the principle of mathematical induction, we have 𝑎_𝑘≦𝐴_𝑘𝑘∈ℕ₀ (3.31) as desired.
By means of Lemma 2.1 and (3.26), we know that 𝐴_𝑘=1(𝑘+1)!^𝑘_𝑗=02𝜆+𝑗𝑘∈ℕ₀. (3.32) Combining (3.31) and (3.32), we readily get the coefficient estimates asserted by Theorem 3.4.
For the sharpness, we consider the function 𝑓 given by (3.14). A simple calculation shows that Re−𝑧𝑓^′(𝑧)𝑓(𝑧)()𝑧=Re1+2𝛼−31−𝑧≧2−𝛼>𝛼. (3.33) Thus, the function 𝑓 belongs to the class ℳ𝒮_𝛼. Since 0<𝛼<1, we have 𝜆=1−𝛼. (3.34) Then 𝑓 becomes 𝑓(𝑧)=𝑧⁻¹(1−𝑧)^−2𝜆=𝑧^−1∞_𝑛=0𝑛()−2𝜆−𝑧^𝑛=1𝑧+^∞_𝑛=02𝜆(2𝜆+1)⋯(2𝜆+𝑛)()!𝑧𝑛+1^𝑛. (3.35) This completes the proof of Theorem 3.4.

Theorem 3.5. If 𝑓∈Σ satisfies the inequality ^∞_𝑘=0(𝑘+|𝑘+2𝛼|)𝑎_𝑘|1≦1−2𝛼−1|𝛼−2<12, (3.36) then 𝑓∈ℳ𝒮_𝛼.

Proof. To prove 𝑓∈ℳ𝒮_𝛼, it suffices to show that 𝑓(𝑧)𝑧𝑓^′(𝑧)+1<12𝛼2|𝛼|(𝑧∈𝕌), (3.37) which is equivalent to 𝑧𝑓^′(𝑧)+2𝛼𝑓(𝑧)𝑧𝑓^′(𝑧)<1𝑧∈𝕌^∗. (3.38) From (3.36), we know that 1−^∞_𝑘=0𝑘𝑎_𝑘≧|2𝛼−1|+^∞_𝑘=0|𝑘+2𝛼|𝑎_𝑘>0. (3.39)
Now, by the maximum modulus principle, we deduce from (1.1) and (3.39) that 𝑧𝑓^′(𝑧)+2𝛼𝑓(𝑧)𝑧𝑓^′(𝑧)=(2𝛼−1)+∑^∞_𝑘=0(𝑘+2𝛼)𝑎_𝑘𝑧^𝑘+1−1+∑^∞_𝑘=0𝑘𝑎_𝑘𝑧^𝑘+1≦|2𝛼−1|+∑^∞_𝑘=0|𝑘+2𝛼|𝑎_𝑘|𝑧|^𝑘+11−∑^∞_𝑘=0𝑘𝑎_𝑘|𝑧|^𝑘+1<|2𝛼−1|+∑^∞_𝑘=0|𝑘+2𝛼|𝑎_𝑘1−∑^∞_𝑘=0𝑘𝑎_𝑘≦1, (3.40) which implies that the assertion of Theorem 3.5 holds.

Theorem 3.6. If 𝑓∈Σ satisfies the condition 1+𝑧𝑓^′′(𝑧)𝑓^′(𝑧)−𝑧𝑓^′(𝑧)𝑓(𝑧)<1−𝛼12𝛼2<𝛼<1, (3.41) then 𝑓∈ℳ𝒮_𝛼.

Proof. Define the function 𝜑 by 𝜑(𝑧)∶=𝑧𝑓^′(𝑧)/𝑓(𝑧)+1𝑧𝑓^′(𝑧)(𝑧)()/𝑓+2𝛼−1𝑧∈𝕌. (3.42) Then we see that 𝜑 is analytic in 𝕌 with 𝜑(0)=0.
It follows from (3.42) that −𝑧𝑓^′(𝑧)𝑓(𝑧)=1+(1−2𝛼)𝜑(𝑧)(𝑧).1−𝜑 (3.43) By differentiating both sides of (3.43) logarithmically, we obtain 1+𝑧𝑓^′′(𝑧)𝑓^′(𝑧)−𝑧𝑓^′(𝑧)𝑓(𝑧)=2(1−𝛼)𝑧𝜑^′(𝑧)()𝜑(𝑧(𝑧).1+1−2𝛼)1−𝜑 (3.44) From (3.41) and (3.44), we find that 1+𝑧𝑓^′′(𝑧)𝑓^′(𝑧)−𝑧𝑓^′(𝑧)𝑓(𝑧)=2(1−𝛼)𝑧𝜑^′(𝑧)()𝜑(𝑧(𝑧)<1+1−2𝛼)1−𝜑1−𝛼.2𝛼 (3.45) Next, we claim that |𝜑(𝑧)|<1. Indeed, if not, there exists a point 𝑧₀∈𝕌 such that max_{|𝑧|≦𝑧0}𝜑(𝑧)=𝜑𝑧₀=1. (3.46) By Lemma 2.2, we have 𝜑𝑧₀=𝑒^𝑖𝜃,𝑧₀𝜑^′𝑧₀=𝑡𝑒^𝑖𝜃(𝑡≧1). (3.47) Moreover, for 𝑧=𝑧₀, we find from (3.44) and (3.47) that 𝑧1+⁰𝑓^′′𝑧₀𝑓^′𝑧₀−𝑧₀𝑓^′𝑧₀𝑓𝑧₀=2(1−𝛼)𝑡𝑒^𝑖𝜃1+(1−2𝛼)𝑒^𝑖𝜃1−𝑒^𝑖𝜃=2(1−𝛼)𝑡1+2(1−2𝛼)cos𝜃+(1−2𝛼)²⋅√≧2−2cos𝜃1−𝛼12𝛼2<𝛼<1. (3.48) But (3.48) contradicts to (3.45). Therefore, we conclude that |𝜑(𝑧)|<1, that is 𝑧𝑓^′(𝑧)/𝑓(𝑧)+1𝑧𝑓^′(𝑧)/𝑓(𝑧)+2𝛼−1<1, (3.49) which shows that 𝑓∈ℳ𝒮_𝛼.

Acknowledgments

The present investigation was supported by the National Natural Science Foundation under Grants 11101053 and 11226088, the Key Project of Chinese Ministry of Education under Grant 211118, the Excellent Youth Foundation of Educational Committee of Hunan Province under Grant 10B002, the Open Fund Project of the Key Research Institute of Philosophies and Social Sciences in Hunan Universities under Grants 11FEFM02 and 12FEFM02, and the Key Project of Natural Science Foundation of Educational Committee of Henan Province under Grant 12A110002 of the People’s Republic of China. The authors would like to thank the referees for their careful reading and valuable suggestions which essentially improved the quality of this paper.