On Certain Subclass of Meromorphic Spirallike Functions Involving the Hypergeometric Function

Abstract

We introduce and investigate a new subclass ℳ¹_𝑚(𝜃,𝜆,𝜂) of meromorphic spirallike functions. Such results as integral representations, convolution properties, and coefficient estimates are proved. The results presented here would provide extensions of those given in earlier works. Several other results are also obtained.

1. Introduction

Let ℳ denote the class of functions 𝑓 of the form 𝑓(𝑧)=1𝑧+^∞_𝑛=1𝑎_𝑛𝑧^𝑛, (1) which are analytic in the punctured open unit disk: 𝕌^∗∶={𝑧∶𝑧∈ℂ,0<|𝑧|<1}=∶𝕌⧵{0}. (2)

Let 𝑓, 𝑔∈ℳ, where 𝑓 is given by (1) and 𝑔 is defined by 𝑔(𝑧)=1𝑧+^∞_𝑛=1𝑏_𝑛𝑧^𝑛. (3) Then the Hadamard product (or convolution) 𝑓∗𝑔 of 𝑓 and 𝑔 is defined by (𝑧)=1𝑓∗𝑔𝑧+^∞_𝑛=1𝑎_𝑛𝑏_𝑛𝑧^𝑛∶=𝑔∗𝑓(𝑧). (4)

Let 𝒫 denote the class of functions 𝑝 given by 𝑝(𝑧)=1+^∞_𝑛=1𝑝_𝑛𝑧^𝑛(𝑧∈𝕌), (5) which are analytic in 𝕌 and satisfy the condition ℜ𝑝(𝑧)>0(𝑧∈𝕌). (6)

For 𝜃 which is real with |𝜃|<𝜋/2, 0≤𝛾<1, we denote by ℳ𝒮^∗(𝜃,𝛾) and ℳ𝒦^∗(𝜃,𝛾) the subclasses of 𝑓∈ℳ which are defined, respectively, by ℜ𝑒^𝑖𝜃𝑧𝑓^′(𝑧)𝑓(𝑧)<−𝛾cos𝜃𝑧∈𝕌^∗,ℜ𝑒^𝑖𝜃𝑧𝑓^′(𝑧)^′𝑓^′(𝑧)<−𝛾cos𝜃𝑧∈𝕌^∗. (7) By setting 𝜃=0 in (7), we get the well-known subclasses of 𝑓∈ℳ consisting of meromorphic functions which are starlike and convex of order 𝛾(0≤𝛾<1), respectively. For some recent investigations on meromorphic spirallike functions and related topics, see, for example, the earlier works [1–4] and the references cited therein.

For 𝜂>1, Wang et al. [5] and Nehari and Netanyahu [6] introduced and studied the subclass ℳ(𝜂) of ℳ consisting of functions 𝑓 satisfying ℜ𝑧𝑓^′(𝑧)𝑓(𝑧)>−𝜂𝑧∈𝕌^∗. (8)

Let 𝒜 be the class of functions of the form 𝑓(𝑧)=𝑧+^∞_𝑛=1𝑎_𝑛𝑧^𝑛, (9) which are analytic in 𝕌. A function 𝑓∈𝒜 is said to be in the class 𝒮^∗(𝜃,𝛿,𝛾) if it satisfies the condition ℜ𝑒^𝑖𝜃𝑧𝑓^′(𝑧)(1−𝛿)𝑓(𝑧)+𝛿𝑧𝑓^′(𝑧)|𝜃|<𝜋>𝛾cos𝜃𝑧∈𝕌;2;0≤𝛿<1;0≤𝛾<1. (10) The function class 𝒮^∗(𝜃,𝛿,𝛾) is introduced and studied recently by Orhan et al. [7]. An analogous of the class 𝒮^∗(𝜃,𝛿,𝛾) has been studied by Murugusundaramoorthy [8].

For complex parameters 𝛼₁,…,𝛼_𝑙 and 𝛽₁,…,𝛽_𝑚(𝛽_𝑗≠0,−1,−2,…;𝑗=1,2,…,𝑚) the generalized hypergeometric function _𝑙𝐹_𝑚(𝑧) is defined by _𝑙𝐹_𝑚(𝑧)≡_𝑙𝐹_𝑚𝛼₁,…,𝛼_𝑙;𝛽₁,…,𝛽_𝑚;𝑧∶=^∞_𝑛=0𝛼_1𝑛⋯𝛼_𝑙𝑛𝛽_1𝑛⋯𝛽_𝑚𝑛𝑧^𝑛𝑛!𝑙≤𝑚+1;𝑙,𝑚∈ℕ₀∶=ℕ∪{0};𝑧∈𝕌, (11) where ℕ denotes the set of all positive integers and (𝑎)_𝑘 is the Pochhammer symbol defined by (𝑎)_𝑛=𝑎()⋯()1,𝑛=0,𝑎+1)(𝑎+2𝑎+𝑛−1,𝑛∈ℕ;𝑎∈ℂ. (12)

For a function 𝑓∈ℳ, we consider a linear operator (which is a meromorphically modified version of the familiar Dziok-Srivastava linear operator [9, 10]: ℋ^𝑙_𝑚𝛼,𝛽𝑓(𝑧)=ℋ𝛼₁,…,𝛼_𝑙;𝛽₁,…,𝛽_𝑚𝑓(𝑧)=𝑧_𝑙⁻¹𝐹_𝑚𝛼₁,…,𝛼_𝑙;𝛽₁,…,𝛽_𝑚(𝑧),;𝑧∗𝑓 (13) where 𝛼_𝑠>0, 𝛽_𝑡>0(𝑠=1,…,𝑙;𝑡=1,…,𝑚;𝑙≤𝑚+1;𝑙,𝑚∈ℕ₀).

From the definition of the operator ℋ^𝑙_𝑚[𝛼,𝛽]𝑓, it is easy to observe that ℋ^𝑙_𝑚𝑓(𝑧)=1𝛼,𝛽𝑧+^∞_𝑛=1𝜙_𝑛𝛼,𝛽;𝑙;𝑚𝑎_𝑛𝑧^𝑛, (14) where 𝜙_𝑛=𝛼𝛼,𝛽;𝑙;𝑚^1𝑛+1⋯𝛼_𝑙𝑛+1𝛽_1𝑛+1⋯𝛽_𝑚𝑛+11(𝑛+1)! (15) is a positive number for all 𝑛∈ℕ.

Recently, Aouf [11], Liu and Srivastava [12], and Raina and Srivastava [13] obtained many interesting results involving the linear operator ℋ^𝑙_𝑚[𝛼,𝛽]. In particular, for 𝑙=2,𝑚=1,𝛼₁=𝑎,𝛽₁=𝑐,𝛼₂=1, (16) we obtain the following linear operator: ℒ(𝑎,𝑐)𝑓(𝑧)∶=ℋ𝛼₁,1;𝛽₁𝑓(𝑧)𝑧∈𝕌^∗, (17) which was introduced and investigated earlier by Liu and Srivastava [14] and was further studied in a subsequent investigation by Srivastava et al. [15]. It should also be remarked that the linear operator ℋ^𝑙_𝑚[𝛼,𝛽] is a generalization of other linear operators considered in many earlier investigations (see, e.g., [16–18]).

Using the operator ℋ^𝑙_𝑚[𝛼,𝛽]𝑓, we introduce the following class of meromorphic functions.

Definition 1. For |𝜃|<𝜋/2, 0≤𝜆<1/2, and 𝜂>1, let ℳ^𝑙_𝑚(𝜃,𝜆,𝜂) denote a subclass of ℳ consisting of functions satisfying the condition that ℜ𝑒^𝑖𝜃(1−2𝜆)𝑧ℋ^𝑙_𝑚𝛼,𝛽𝑓(𝑧)^′(1−𝜆)ℋ^𝑙_𝑚𝛼,𝛽𝑓(𝑧)+𝜆𝑧ℋ^𝑙_𝑚𝛼,𝛽𝑓(𝑧)^′>−𝜂cos𝜃,𝑧∈𝕌^∗, (18) where ℋ^𝑙_𝑚[𝛼,𝛽]𝑓 is given by (13).

We note that, for 𝑙=2, 𝑚=1, 𝛼₁=𝛼₂=1, 𝛽₁=1, and 𝜃=𝜆=0, the class ℳ²₁(0,0,𝜂) becomes the class ℳ(𝜂).

In the present paper, we aim at proving some interesting properties such as integral representations, convolution properties, and coefficient estimates for the class ℳ^𝑙_𝑚(𝜃,𝜆,𝜂).

The following lemma will be required in our investigation.

Lemma 2. Suppose that the sequence {𝐴_𝑛}^∞_𝑛=1 is defined by 𝐴₁=(1−2𝜆)𝜂−1cos𝜃(1−𝜆)𝜙₁,𝐴𝛼,𝛽;𝑙;𝑚_𝑛+1=2𝜂−1cos𝜃(𝑛+2)(1−𝜆)𝜙_𝑛+1×𝛼,𝛽;𝑙;𝑚1−2𝜆+^𝑛_𝑘=1𝜙_𝑘𝛼,𝛽;𝑙;𝑚(1−𝜆+𝑘𝜆)𝐴_𝑘. (19) Then 𝐴_𝑛=(1−2𝜆)𝜂−1cos𝜃(1−𝜆)^𝑛𝜙_𝑛×𝛼,𝛽;𝑙;𝑚_{𝑛−1𝑘=1}(𝑘+1)(1−𝜆)+2(1−𝜆+𝑘𝜆)𝜂−1cos𝜃,().𝑘+2𝑛≥2 (20)

Proof. From (19), we have (𝑛+2)(1−𝜆)𝜙_𝑛+1𝛼,𝛽;𝑙;𝑚𝐴_𝑛+1+=2𝜂−1cos𝜃1−2𝜆_𝑛𝑘=1𝜙_𝑘𝛼,𝛽;𝑙;𝑚(1−𝜆+𝑘𝜆)𝐴_𝑘,(𝑛+1)(1−𝜆)𝜙_𝑛𝛼,𝛽;𝑙;𝑚𝐴_𝑛+=2𝜂−1cos𝜃1−2𝜆_{𝑛−1𝑘=1}𝜙_𝑘𝛼,𝛽;𝑙;𝑚(1−𝜆+𝑘𝜆)𝐴_𝑘. (21) Combining (21), we find that 𝐴_𝑛+1𝐴_𝑛=(𝑛+1)(1−𝜆)+2(1−𝜆+𝑛𝜆)𝜂−1cos𝜃()⋅𝜙𝑛+2)(1−𝜆_𝑛𝛼,𝛽;𝑙;𝑚𝜙_𝑛+1.𝛼,𝛽;𝑙;𝑚 (22) Thus, for 𝑛≥2, we deduce from (22) that 𝐴_𝑛=𝐴_𝑛𝐴_𝑛−1⋯𝐴₃𝐴₂⋅𝐴₂𝐴₁⋅𝐴₁=𝑛(1−𝜆)+2(1−𝑛𝜆)𝜂−1cos𝜃()⋯×3()()𝑛+1)(1−𝜆1−𝜆+21+𝜆𝜂−1cos𝜃4()⋅2()1−𝜆1−𝜆+2𝜂−1cos𝜃3()⋅𝜙1−𝜆_𝑛−1𝛼,𝛽;𝑙;𝑚𝜙_𝑛⋯𝜙𝛼,𝛽;𝑙;𝑚²𝛼,𝛽;𝑙;𝑚𝜙₃⋅𝜙𝛼,𝛽;𝑙;𝑚₁𝛼,𝛽;𝑙;𝑚𝜙₂⋅()𝛼,𝛽;𝑙;𝑚1−2𝜆𝜂−1cos𝜃(1−𝜆)𝜙₁=()𝛼,𝛽;𝑙;𝑚1−2𝜆𝜂−1cos𝜃(1−𝜆)^𝑛𝜙_𝑛×𝛼,𝛽;𝑙;𝑚_{𝑛−1𝑘=1}(𝑘+1)(1−𝜆)+2(1−𝜆+𝑘𝜆)𝜂−1cos𝜃.𝑘+2 (23) This completes the proof of Lemma 2.

2. Main Results

We begin by proving the following integral representation for the class ℳ^𝑙_𝑚(𝜃,𝜆,𝜂).

Theorem 3. Let 𝑓∈ℳ^𝑙_𝑚(𝜃,𝜆,𝜂). Then 𝑓(𝑧)=𝑧⁻¹+^∞_𝑛=1𝜙_𝑛⁻¹𝛼,𝛽;𝑙,𝑚𝑧^𝑛∗𝑧⁻¹⋅exp^𝑧₀()×1−2𝜆𝜂−11+𝑒^−2𝑖𝜃𝜔(𝑡)×[(1−𝜆)(1−𝜔(𝑡))−𝜆𝜂−11+𝑒^−2𝑖𝜃×𝜔(𝑡)]𝑡⁻¹𝑑𝑡,𝑧∈𝕌^∗, (24) where 𝜔 is analytic in 𝕌 with 𝜔(0)=0 and |𝜔(𝑧)|<1.

Proof. Suppose that 𝑓∈ℳ^𝑙_𝑚(𝜃,𝜆,𝜂) and 𝜏(𝑧)∶=𝑒^𝑖𝜃(1−2𝜆)𝑧ℋ^𝑙_𝑚𝛼,𝛽𝑓(𝑧)^′×(1−𝜆)ℋ^𝑙_𝑚𝑓(𝑧)ℋ𝛼,𝛽+𝜆𝑧^𝑙_𝑚𝛼,𝛽𝑓(𝑧)^′−1+𝜂cos𝜃+𝑖sin𝜃×𝜂−1cos𝜃⁻¹,(𝑧∈𝕌). (25) We know that 𝜏∈𝒫, which implies 𝑒^𝑖𝜃(1−2𝜆)𝑧ℋ^𝑙_𝑚𝛼,𝛽𝑓(𝑧)^′×(1−𝜆)ℋ^𝑙_𝑚𝛼,𝛽𝑓(𝑧)+𝜆𝑧ℋ^𝑙_𝑚𝛼,𝛽𝑓(𝑧)^′−1+𝜂cos𝜃+𝑖sin𝜃×𝜂−1cos𝜃⁻¹=1+𝜔(𝑧)(𝑧),1−𝜔 (26) where 𝜔 is analytic in 𝕌 with 𝜔(0)=0 and |𝜔(𝑧)|<1. We find from (26) that (1−2𝜆)𝑧ℋ^𝑙_𝑚𝛼,𝛽𝑓(𝑧)^′(1−𝜆)ℋ^𝑙_𝑚𝛼,𝛽𝑓(𝑧)+𝜆𝑧ℋ^𝑙_𝑚𝛼,𝛽𝑓(𝑧)^′=−1+1+𝜂−11+𝑒^−2𝑖𝜃𝜔(𝑧)(𝑧),1−𝜔 (27) which follows ℋ^𝑙_𝑚𝛼,𝛽𝑓(𝑧)^′ℋ^𝑙_𝑚𝑓(𝑧)+1𝛼,𝛽𝑧=(1−2𝜆)𝜂−11+𝑒^−2𝑖𝜃𝜔(𝑧)𝑧(1−𝜆)(1−𝜔(𝑧))−𝜆𝜂−11+𝑒^−2𝑖𝜃𝜔(𝑧.) (28) Integrating both sides of (28) yields log𝑧ℋ^𝑙_𝑚𝑓(𝑧)=𝛼,𝛽_𝑧0(1−2𝜆)𝜂−11+𝑒^−2𝑖𝜃𝜔(𝑡)(1−𝜆)(1−𝜔(𝑡))−𝜆𝜂−11+𝑒^−2𝑖𝜃𝜔(𝑡)𝑡𝑑𝑡. (29) From (29), we obtain ℋ^𝑙_𝑚𝛼,𝛽𝑓(𝑧)=𝑧⁻¹⋅exp^𝑧₀(1−2𝜆)1+𝑒^−2𝑖𝜃𝜔(𝑡)×(𝑡×𝜂−1(1−𝜆)(1−𝜔))−𝜆𝜂−11+𝑒^−2𝑖𝜃𝜔(𝑡)𝑡⁻¹𝑑𝑡. (30) Thus, the assertion (24) of Theorem 3 follows directly from (30).

Next, we derive a convolution property for the class ℳ^𝑙_𝑚(𝜃,𝜆,𝜂).

Theorem 4. Let 𝜉∈ℂ and |𝜉|=1. Then 𝑓∈ℳ^𝑙_𝑚(𝜃,𝜆,𝜂) if and only if )⋅1𝑓∗(1−𝜉𝑧−^∞_𝑛=1𝑛𝜙_𝑛𝛼,𝛽;𝑙;𝑚𝑧^𝑛+𝜉𝜂+𝜉𝜂−1𝑒^−2𝑖𝜃⋅1−1𝑧+^∞_𝑛=11−𝜆+𝑛𝜆𝜙1−2𝜆^𝑛𝛼,𝛽;𝑙;𝑚𝑧^𝑛≠0,𝑧∈𝕌^∗. (31)

Proof. From the definition (18), we know that 𝑓∈ℳ^𝑙_𝑚(𝜃,𝜆,𝜂) if and only if 𝑒^𝑖𝜃(1−2𝜆)𝑧ℋ^𝑙_𝑚𝛼,𝛽𝑓(𝑧)^′×(1−𝜆)ℋ^𝑙_𝑚𝛼,𝛽𝑓(𝑧)+𝜆𝑧ℋ^𝑙_𝑚𝛼,𝛽𝑓(𝑧)^′−1+𝜂cos𝜃+𝑖sin𝜃×𝜂−1cos𝜃⁻¹≠1+𝜉,1−𝜉𝑧∈𝕌^∗;𝜉=1, (32) which is equivalent to −(1−𝜉)𝑧ℋ^𝑙_𝑚𝛼,𝛽𝑓(𝑧)^′+𝜉𝜂+𝜉𝜂−1𝑒^−2𝑖𝜃⋅()ℋ−11−𝜆^𝑙_𝑚𝛼,𝛽𝑓(𝑧)+𝜆𝑧ℋ^𝑙_𝑚𝛼,𝛽𝑓(𝑧)^′1−2𝜆≠0. (33) On the other hand, we find from (14) that −𝑧ℋ^𝑙_𝑚𝛼,𝛽𝑓(𝑧)^′=1𝑧−^∞_𝑛=1𝑛𝜙_𝑛𝛼,𝛽;𝑙;𝑚𝑎_𝑛𝑧^𝑛(𝑧)∗1=𝑓𝑧−^∞_𝑛=1𝑛𝜙_𝑛𝛼,𝛽;𝑙;𝑚𝑧^𝑛,(1−𝜆)ℋ^𝑙_𝑚𝛼,𝛽𝑓(𝑧)+𝜆𝑧ℋ^𝑙_𝑚𝛼,𝛽𝑓(𝑧)^′=11−2𝜆𝑧+^∞_𝑛=11−𝜆+𝑛𝜆𝜙1−2𝜆^𝑛𝛼,𝛽;𝑙;𝑚𝑎_𝑛𝑧^𝑛(𝑧)∗1=𝑓𝑧+^∞_𝑛=11−𝜆+𝑛𝜆𝜙1−2𝜆^𝑛𝛼,𝛽;𝑙;𝑚𝑧^𝑛. (34) Combining (33) and (34), we get assertion (31) of Theorem 4.

Now, we discuss the coefficient estimates for functions in the class ℳ^𝑙_𝑚(𝜃,𝜆,𝜂).

Theorem 5. Suppose that 𝑓∈ℳ^𝑙_𝑚(𝜃,𝜆,𝜂). Then 𝑎₁≤(1−2𝜆)𝜂−1cos𝜃(1−𝜆)𝜙₁,𝑎𝛼,𝛽;𝑙;𝑚_𝑛≤(1−2𝜆)𝜂−1cos𝜃(1−𝜆)^𝑛𝜙_𝑛×𝛼,𝛽;𝑙;𝑚_{𝑛−1𝑘=1}(𝑘+1)(1−𝜆)+2(1−𝜆+𝑘𝜆)𝜂−1cos𝜃,({1.𝑘+2𝑛∈ℕ⧵}) (35)

Proof. Let 𝑓∈ℳ^𝑙_𝑚(𝜃,𝜆,𝜂). Then there exists 𝜏∈𝒫 such that 𝑒^𝑖𝜃(1−2𝜆)𝑧ℋ^𝑙_𝑚𝛼,𝛽𝑓(𝑧)^′(1−𝜆)ℋ^𝑙_𝑚𝛼,𝛽𝑓(𝑧)+𝜆𝑧ℋ^𝑙_𝑚𝛼,𝛽𝑓(𝑧)^′=𝜂−1cos𝜃𝜏(𝑧)−𝜂cos𝜃−𝑖sin𝜃,𝑧∈𝕌^∗. (36) It follows from (36) that 𝑒^𝑖𝜃(1−2𝜆)𝑧ℋ^𝑙_𝑚𝛼,𝛽𝑓(𝑧)^′=(1−𝜆)ℋ^𝑙_𝑚𝛼,𝛽𝑓(𝑧)+𝜆𝑧ℋ^𝑙_𝑚𝛼,𝛽𝑓(𝑧)^′×𝜂−1cos𝜃𝜏(𝑧)−𝜂cos𝜃−𝑖sin𝜃. (37) Combining (1) and (37), we have 𝑒^𝑖𝜃(−11−2𝜆)𝑧+^∞_𝑛=1𝑛𝜙_𝑛𝛼,𝛽;𝑙;𝑚𝑎_𝑛𝑧^𝑛=)1(1−2𝜆𝑧+^∞_𝑛=1(1−𝜆+𝑛𝜆)𝜙_𝑛𝛼,𝛽;𝑙;𝑚𝑎_𝑛𝑧^𝑛⋅−𝑒^𝑖𝜃+𝜂−1cos𝜃𝜏₁𝑧+𝜂−1cos𝜃𝜏₂𝑧²+⋯. (38) Evaluating the coefficient of 𝑧^𝑛 in both sides of (38) yields 2𝑒^𝑖𝜃(1−𝜆)𝜙₁𝛼,𝛽;𝑙;𝑚𝑎₁=(1−2𝜆)𝜂−1cos𝜃𝜏₂,𝑒^𝑖𝜃(𝑛+1)(1−𝜆)𝜙_𝑛𝛼,𝛽;𝑙;𝑚𝑎_𝑛=𝜂−1cos𝜃(1−2𝜆)𝜏_𝑛+1+^𝑛−1_𝑘=1𝜙_𝑘𝛼,𝛽;𝑙;𝑚(1−𝜆+𝑘𝜆)𝑎_𝑘𝜏_𝑛−𝑘,(𝑛≥2). (39) By observing the fact that |𝜏_𝑛|≤2 for 𝑛∈ℕ, we find from (39) that 𝑎₁≤(1−2𝜆)𝜂−1cos𝜃(1−𝜆)𝜙₁,𝛼,𝛽;𝑙;𝑚𝑎_𝑛≤2𝜂−1cos𝜃(𝑛+1)(1−𝜆)𝜙_𝑛×𝛼,𝛽;𝑙;𝑚1−2𝜆+^𝑛−1_𝑘=1𝜙_𝑘𝛼,𝛽;𝑙;𝑚(1−𝜆+𝑘𝜆)𝑎_𝑘,(𝑛≥2). (40)(41) Now we define the sequence {𝐴_𝑛}^∞_𝑛=1 as follows: 𝐴₁=(1−2𝜆)𝜂−1cos𝜃(1−𝜆)𝜙₁,𝐴𝛼,𝛽;𝑙;𝑚_𝑛+1=2𝜂−1cos𝜃(𝑛+2)(1−𝜆)𝜙_𝑛+1×𝛼,𝛽;𝑙;𝑚1−2𝜆+^𝑛_𝑘=1𝜙_𝑘𝛼,𝛽;𝑙;𝑚(1−𝜆+𝑘𝜆)𝐴_𝑘. (42) In order to prove that 𝑎_𝑛≤𝐴_𝑛(𝑛∈ℕ), (43) we use the principle of mathematical induction. Note that 𝑎₁≤𝐴₁=(1−2𝜆)𝜂−1cos𝜃(1−𝜆)𝜙₁.𝛼,𝛽;𝑙;𝑚 (44) Therefore, assume that 𝑎_𝑘≤𝐴_𝑘(𝑘=1,2,…,𝑛;𝑛∈ℕ). (45) Combining (41) and (42), we get 𝑎_𝑛+1≤2𝜂−1cos𝜃(𝑛+2)(1−𝜆)𝜙_𝑛+1×𝛼,𝛽;𝑙;𝑚1−2𝜆+^𝑛_𝑘=1𝜙_𝑘𝛼,𝛽;𝑙;𝑚(1−𝜆+𝑘𝜆)𝑎_𝑘≤2𝜂−1cos𝜃(𝑛+2)(1−𝜆)𝜙_𝑛+1×𝛼,𝛽;𝑙;𝑚1−2𝜆+^𝑛_𝑘=1𝜙_𝑘𝛼,𝛽;𝑙;𝑚(1−𝜆+𝑘𝜆)𝐴_𝑘=𝐴_𝑛+1. (46) Hence, by the principle of mathematical induction, we have 𝑎_𝑛≤𝐴_𝑛(𝑛∈ℕ), (47) as desired. By means of Lemma 2 and (42), we know that (20) holds. Combining (47) and (20), we readily get the coefficient estimates asserted by Theorem 5.

Remark 6. By setting 𝜃=0, 𝑙=2, 𝑚=1, 𝛼₁=𝛼₂=𝛽₁=1, and 𝜆=0 in Theorem 5, we get the corresponding result due to Wang et al. [5].

In what follows, we present some sufficient conditions for functions belonging to the class ℳ^𝑙_𝑚(𝜃,𝜆,𝜂).

Theorem 7. Let 𝜁 be a real number with 0≤𝜁<1. If 𝑓∈ℳ satisfies the condition (1−2𝜆)𝑧ℋ^𝑙_𝑚𝛼,𝛽𝑓(𝑧)^′(1−𝜆)ℋ^𝑙_𝑚𝑓(𝑧)+𝜆𝑧ℋ^𝑙_𝑚𝛼,𝛽𝑓(𝑧)^′+1≤1−𝜁,𝑧∈𝕌^∗, (48) then 𝑓∈ℳ^𝑙_𝑚(𝜃,𝜆,𝜂) provided that cos𝜃≥1−𝜁.𝜂−1 (49)

Proof. From (48), it follows that (1−2𝜆)𝑧ℋ^𝑙_𝑚𝛼,𝛽𝑓(𝑧)^′(1−𝜆)ℋ^𝑙_𝑚𝛼,𝛽𝑓(𝑧)+𝜆𝑧ℋ^𝑙_𝑚𝛼,𝛽𝑓(𝑧)^′=−1+(1−𝜁)𝜔(𝑧), (50) where 𝜔 is analytic in 𝕌 with 𝜔(0)=0 and |𝜔(𝑧)|<1. Thus, we have ℜ𝑒^𝑖𝜃(1−2𝜆)𝑧ℋ^𝑙_𝑚𝛼,𝛽𝑓(𝑧)^′(1−𝜆)ℋ^𝑙_𝑚𝛼,𝛽𝑓(𝑧)−𝜆𝑧ℋ^𝑙_𝑚𝛼,𝛽𝑓(𝑧)^′=ℜ𝑒^𝑖𝜃(()𝜔(𝑧()ℜ𝑒−1+1−𝜁))=−cos𝜃+1−𝜁^𝑖𝜃𝜔(𝑧(𝜔(𝑧())≥−cos𝜃−1−𝜁)|)|>−cos𝜃−1−𝜁≥−𝜂cos𝜃, (51) provided that cos𝜃≥(1−𝜁)/(𝜂−1). This completes the proof of Theorem 7.

If we take 𝜁=1−(𝜂−1)cos𝜃 in Theorem 7, we obtain the following result.

Corollary 8. If 𝑓∈ℳ satisfies the inequality (1−2𝜆)𝑧ℋ^𝑙_𝑚𝛼,𝛽𝑓(𝑧)^′(1−𝜆)ℋ^𝑙_𝑚𝛼,𝛽𝑓(𝑧)+𝜆𝑧ℋ^𝑙_𝑚𝛼,𝛽𝑓(𝑧)^′≤+1𝜂−1cos𝜃, (52) then 𝑓∈ℳ^𝑙_𝑚(𝜃,𝜆,𝜂).

Theorem 9. If a function 𝑓∈ℳ given by (1) satisfies the inequality (1−𝜆)(1+𝑛)^∞_𝑛=1sec𝜃+𝜂−1(1−𝜆+𝑛𝜆)×𝜙_𝑛𝛼,𝛽;𝑙,𝑚𝑎_𝑛≤(1−2𝜆)𝜂−1, (53) then it belongs to the class ℳ^𝑙_𝑚(𝜃,𝜆,𝜂).

Proof. In virtue of Corollary 8, it suffices to show that condition (52) holds. We observe that (1−2𝜆)𝑧ℋ^𝑙_𝑚𝛼,𝛽𝑓(𝑧)^′(1−𝜆)ℋ^𝑙_𝑚𝛼,𝛽𝑓(𝑧)+𝜆𝑧ℋ^𝑙_𝑚𝛼,𝛽𝑓(𝑧)^′=()×∑+11−𝜆_∞𝑛=1(1+𝑛)𝜙_𝑛𝛼,𝛽;𝑙,𝑚𝑎_𝑛𝑧^𝑛+11−2𝜆+∑^∞_𝑛=1(1−𝜆+𝑛𝜆)𝜙_𝑛𝛼,𝛽;𝑙,𝑚𝑎_𝑛𝑧^𝑛+1<(1−𝜆)∑^∞_𝑛=1(1+𝑛)𝜙_𝑛𝛼,𝛽;𝑙,𝑚𝑎_𝑛1−2𝜆−∑^∞_𝑛=1(1−𝜆+𝑛𝜆)𝜙_𝑛𝛼,𝛽;𝑙,𝑚𝑎_𝑛. (54) The last expression is bounded by (𝜂−1)cos𝜃, if (1−𝜆)^∞_𝑛=1(1+𝑛)𝜙_𝑛𝛼,𝛽;𝑙,𝑚𝑎_𝑛≤−𝜂−1cos𝜃1−2𝜆_∞𝑛=1(1−𝜆+𝑛𝜆)𝜙_𝑛𝛼,𝛽;𝑙,𝑚𝑎_𝑛, (55) which is equivalent to (1−𝜆)(1+𝑛)^∞_𝑛=1sec𝜃+𝜂−1(1−𝜆+𝑛𝜆)×𝜙_𝑛𝛼,𝛽;𝑙,𝑚𝑎_𝑛≤(1−2𝜆)𝜂−1. (56) This completes the proof of Theorem 9.

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 Grant nos. 11301008 and 11226088, 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.