On Certain Class of Analytic Functions Related to Cho-Kwon-Srivastava Operator

Abstract

Motivated by a multiplier transformation and some subclasses of meromorphic functions which were defined by means of the Hadamard product of the Cho-Kwon-Srivastava operator, we define here a similar transformation by means of the Ghanim and Darus operator. A class related to this transformation will be introduced and the properties will be discussed.

1. Introduction

Let Σ denote the class of meromorphic functions 𝑓(𝑧) normalized by𝑓(𝑧)=1𝑧+^∞_𝑛=1𝑎_𝑛𝑧^𝑛, (1.1) which are analytic in the punctured unit disk 𝑈={𝑧∶0<|𝑧|<1}. For 0≤𝛽, we denote by 𝑆^∗(𝛽) and 𝑘(𝛽) the subclasses of Σ consisting of all meromorphic functions which are, respectively, starlike of order 𝛽 and convex of order 𝛽 in 𝑈 (cf. e.g., [1–4]).

For functions 𝑓_𝑗(𝑧)(𝑗=1;2) defined by𝑓_𝑗(𝑧)=1𝑧+^∞_𝑛=1𝑎_𝑛,𝑗𝑧^𝑛, (1.2) we denote the Hadamard product (or convolution) of 𝑓₁(𝑧) and 𝑓₂(𝑧) by𝑓₁∗𝑓₂=1𝑧+^∞_𝑛=1𝑎_𝑛,1𝑎_𝑛,2𝑧^𝑛. (1.3) Let us define the function 𝜙(𝛼,𝛽;𝑧) by𝜙=1𝛼,𝛽;𝑧𝑧+^∞_𝑛=0(𝛼)_𝑛+1𝛽_𝑛+1𝑧^𝑛, (1.4) for 𝛽≠0,−1,−2,…, and 𝛼∈ℂ/{0}, where (𝜆)𝑛=𝜆(𝜆+1)_𝑛+1 is the Pochhammer symbol. We note that 𝜙=1𝛼,𝛽;𝑧𝑧²𝐹₁1,𝛼,𝛽;𝑧, (1.5) where ₂𝐹₁𝑏,𝛼,𝛽;𝑧=^∞_𝑛=0(𝑏)_𝑛(𝛼)_𝑛𝛽_𝑛𝑧^𝑛𝑛! (1.6) is the well-known Gaussian hypergeometric function.

Let us put 𝑞_𝜆,𝜇(𝑧)=1𝑧+^∞_𝑛=1𝜆𝑛+1+𝜆^𝜇𝑧^𝑛,𝜆>0,𝜇≥0. (1.7) Corresponding to the functions 𝜙(𝛼,𝛽;𝑧) and 𝑞_𝜆,𝜇(𝑧) and using the Hadamard product for 𝑓(𝑧)∈Σ, we define a new linear operator 𝐿(𝛼,𝛽,𝜆,𝜇)𝑓(𝑧) on ∑ by 𝐿𝛼,𝛽,𝜆,𝜇𝑓(𝑧)=𝑓(𝑧)∗𝜙𝛼,𝛽;𝑧∗𝑞_𝜆,𝜇(𝑧=1)𝑧+^∞_𝑛=1(𝛼)_𝑛+1𝛽_𝑛+1𝜆𝑛+1+𝜆^𝜇𝑎_𝑛𝑧^𝑛. (1.8) The meromorphic functions with the generalized hypergeometric functions were considered recently by Dziok and Srivastava [5, 6], Liu [7], Liu and Srivastava [8–10], and Cho and Kim [11].

For a function 𝑓∈𝐿(𝛼,𝛽,𝜆,𝜇)𝑓(𝑧), we define 𝐼^𝜇,0_{𝛼,𝛽,𝜆}=𝐿𝛼,𝛽,𝜆,𝜇𝑓(𝑧), (1.9) and, for 𝑘=1,2,3,…,𝐼^𝜇,𝑘_{𝛼,𝛽,𝜆}𝑓(𝑧)=𝑧𝐼^𝑘−1𝐿𝛼,𝛽,𝜆,𝜇𝑓(𝑧)^′+2𝑧=1𝑧+^∞_𝑛=1𝑛^𝑘(𝛼)_𝑛+1𝛽_𝑛+1𝜆𝑛+1+𝜆^𝜇𝑎_𝑛𝑧^𝑛. (1.10) Note that if 𝑛=𝛽, 𝑘=0, the operator 𝐼^𝜇,0_{𝛼,𝑛,𝜆} reduced to the one introduced by Cho et al. [12] for 𝜇∈ℕ₀=ℕ∪0. It was known that the definition of the operator 𝐼^𝜇,0_{𝛼,𝑛,𝜆} was motivated essentially by the Choi-Saigo-Srivastava operator [13] for analytic functions, which includes a simpler integral operator studied earlier by Noor [14] and others (cf. [15–17]). Note also the operator 𝐼^0,𝑘_𝛼,𝛽 has been recently introduced and studied by Ghanim and Darus [18] and Ghanim et al. [19], respectively. To our best knowledge, the recent work regarding operator 𝐼^𝜇,0_{𝛼,𝑛,𝜆} was charmingly studied by Piejko and Sokól [20]. Moreover, the operator 𝐼^𝜇,𝑘_{𝛼,𝛽,𝜆} was then defined and studied by Ghanim and Darus [21]. In the same direction, we will study for the operator 𝐼^𝜇,𝑘_{𝛼,𝛽,𝜆} given in (1.10).

Now, it follows from (1.8) and (1.10) that𝑧𝐼^𝜇,𝑘_{𝛼,𝛽,𝜆}𝑓(𝑧)^′=𝛼𝐼^𝜇,𝑘_{𝛼+1,𝛽,𝜆}𝑓(𝑧)−(𝛼+1)𝐼^𝜇,𝑘_{𝛼,𝛽,𝜆}𝑓(𝑧). (1.11) Making use of the operator 𝐼^𝜇,𝑘_{𝛼,𝛽,𝜆}𝑓(𝑧), we say that a function 𝑓(𝑧)∈Σ is in the class Σ^𝜇,𝑘_{𝛼,𝛽,𝜆}(𝐴,𝐵) if it satisfies the following subordination condition:𝑧𝐼^𝜇,𝑘_{𝛼,𝛽,𝜆}𝑓(𝑧)^′𝐼^𝜇,𝑘_{𝛼,𝛽,𝜆}𝑓(𝑧)()𝑤(𝑧)≺1−𝐵−𝐴1+𝐵𝑤(𝑧),𝑧∈𝑈;−1≤𝐵<𝐴≤1. (1.12) Furthermore, we say that a function 𝑓(𝑧)∈Σ^𝜇,𝑘,+_{𝛼,𝛽,𝜆}(𝐴,𝐵) is a subclass of the class Σ^𝜇,𝑘_{𝛼,𝛽,𝜆}(𝐴,𝐵) of the form𝑓(𝑧)=1𝑧+^∞_𝑛=1𝑎_𝑛𝑧^𝑛𝑎_𝑛>0,𝑧∈𝑈. (1.13) The main object of this paper is to present several inclusion relations and other properties of functions in the classes Σ^𝜇,𝑘_{𝛼,𝛽,𝜆}(𝐴,𝐵) and Σ^𝜇,𝑘,+_{𝛼,𝛽,𝜆}(𝐴,𝐵) which we have introduced here.

2. Main Results

We begin by recalling the following result (popularly known as Jack's Lemma), which we will apply in proving our first inclusion theorem.

Lemma 2.1 (see [Jack's Lemma] [22]). Let the (nonconstant) function 𝑤(𝑧) be analytic in 𝑈 with 𝑤(0)=0. If |𝑤(𝑧)| attains its maximum value on the circle |𝑧|=𝑟<1 at a point 𝑧₀∈𝑈, then 𝑧₀𝑤^′𝑧₀=𝛾𝑤𝑧₀, (2.1) where 𝛾 is a real number and 𝛾≥1.

Theorem 2.2. If ()𝛼>𝐴−𝐵(),1+𝐵−1<𝐵<𝐴≤1 (2.2) then Σ^𝜇,𝑘_{𝛼+1,𝛽,𝜆}(𝐴,𝐵)⊂Σ^𝜇,𝑘_{𝛼,𝛽,𝜆}(𝐴,𝐵). (2.3)

Proof. Let 𝑓∈Σ^𝜇,𝑘_{𝛼+1,𝛽,𝜆}(𝐴,𝐵), and suppose that 𝑧𝐼^𝜇,𝑘_{𝛼,𝛽,𝜆}𝑓(𝑧)^′𝐼^𝜇,𝑘_{𝛼,𝛽,𝜆}𝑓(𝑧)()𝑤(𝑧)=1−𝐵−𝐴(𝑧),1+𝐵𝑤 (2.4) where the function 𝑤(𝑧) is either analytic or meromorphic in 𝑈, with 𝑤(0)=0. By using (2.4) and (1.11), we have 𝛼𝐼^𝜇,𝑘_{𝛼+1,𝛽,𝜆}𝑓(𝑧)𝐼^𝜇,𝑘_{𝛼,𝛽,𝜆}𝑓(𝑧)=𝛼+[𝛼𝐵−(𝐴−𝐵)]𝑤(𝑧)1+𝐵𝑤(𝑧). (2.5) Upon differentiating both sides of (2.5) with respect to 𝑧 logarithmically and using the identity (1.11), we obtain 𝑧𝐼^𝜇,𝑘_{𝛼+1,𝛽,𝜆}𝑓(𝑧)^′𝐼^𝜇,𝑘_{𝛼,𝛽,𝜆}𝑓(𝑧)()𝑤(𝑧)=1−𝐵−𝐴(𝑧)−()1+𝐵𝑤𝐴−𝐵𝑧𝑤^′(𝑧)[(𝑧[(𝑤(𝑧.1+𝐵𝑤)](𝛼+𝛼𝐵−𝐴−𝐵)])) (2.6) We suppose now that max_{|𝑧|≤𝑧0}|𝑤(𝑧)|=𝑤𝑧₀=1(𝑧∈𝑈) (2.7) and apply Jack's Lemma, we thus find that 𝑧₀𝑤^′𝑧₀=𝛾𝑤𝑧₀𝛾≥1. (2.8) By writing 𝑤𝑧₀=𝑒^𝑖𝜃(0≤𝜃<2𝜋) (2.9) and setting 𝑧=𝑧₀ in (2.6), we find after some computations that 𝑧₀𝐼^𝜇,𝑘_{𝛼+1,𝛽,𝜆}𝑓𝑧₀^′+𝐼^𝜇,𝑘_{𝛼+1,𝛽,𝜆}𝑓𝑧₀𝐵𝑧₀𝐼^𝜇,𝑘_{𝛼+1,𝛽,𝜆}𝑓𝑧₀^′+𝐴𝐼^𝜇,𝑘_{𝛼+1,𝛽,𝜆}𝑓𝑧₀²+[(𝑒−1=𝛼+𝛾𝛼𝐵−𝐴−𝐵)]^𝑖𝜃𝛼+𝛼𝐵−𝛾−(𝐴−𝐵)𝑒^𝑖𝜃2=(𝛼()−(−12𝛾1+cos𝜃)[𝐵+1𝐴−𝐵)]𝛼+𝛼𝐵−𝛾−(𝐴−𝐵)𝑒^𝑖𝜃2. (2.10) Set 𝑔(𝜃)=2𝛾(1+cos𝜃)[𝛼(𝐵+1)−(𝐴−𝐵)]. (2.11) Then, by hypothesis, we have 𝑔(0)[𝛼()−(𝑔(𝜋)=4𝛾𝐵+1𝐴−𝐵)]≥0,=0, (2.12) which, together, imply that 𝑔(𝜃)≥0(0≤𝜃<2𝜋). (2.13) View of (2.13) and (2.10) would obviously contradict our hypothesis that 𝑓∈Σ^𝜇,𝑘_{𝛼+1,𝛽,𝜆}(𝐴,𝐵). (2.14) Hence, we must have |𝑤(𝑧)|<1(𝑧∈𝑈), (2.15) and we conclude from (2.4) that 𝑓∈Σ^𝜇,𝑘_{𝛼,𝛽,𝜆}(𝐴,𝐵). (2.16) The proof of Theorem 2.2 is thus complete.

3. Properties of the Class 𝑓∈Σ^𝜇,𝑘,+_{𝛼,𝛽,𝜆}(𝐴,𝐵)

Throughout this section, we assume further that 𝛼,𝛽>0 and 𝐴+𝐵≤0(−1<𝐵<𝐴≤1). (3.1) We first determine a necessary and sufficient condition for a function 𝑓∈Σ of the form (1.13) to be in the class 𝑓∈Σ^𝜇,𝑘,+_{𝛼,𝛽,𝜆}(𝐴,𝐵) of meromorphically univalent functions with positive coefficients.

Theorem 3.1. Let 𝑓∈Σ be given by (1.13). Then 𝑓∈Σ^𝜇,𝑘,+_{𝛼,𝛽,𝜆}(𝐴,𝐵) if and only if ^∞_𝑛=1𝑛^𝑘[𝑛()+(𝛼)1−𝐵1−𝐴)](_𝑛+1𝛽_n+1𝜆𝑛+1+𝜆^𝜇𝑎_𝑛≤𝐴−𝐵, (3.2) where, for convenience, the result is sharp for the function 𝑓(𝑧) given by 𝑓(𝑧)=1𝑧+(𝐴−𝐵)(𝑛+1+𝜆)^𝜇𝛽_𝑛+1𝑛^𝑘𝜆^𝜇[𝑛(1−𝐵)+(1−𝐴)](𝛼)_𝑛+1𝑧^𝑛, (3.3) for all 𝑧≠0.

Proof. Suppose that the function 𝑓∈Σ is given by (1.13) and is in the class Σ^𝜇,𝑘,+_{𝛼,𝛽,𝜆}(𝐴,𝐵). Then, from (1.13) and (1.12), we find that 𝑧𝐼^𝜇,𝑘_{𝛼,𝛽,𝜆}𝑓(𝑧)^′+𝐼^𝜇,𝑘_{𝛼,𝛽,𝜆}𝑓(𝑧)𝐵𝑧𝐼^𝜇,𝑘_{𝛼,𝛽,𝜆}𝑓(𝑧)^′+𝐴𝐼^𝜇,𝑘_{𝛼,𝛽,𝜆}𝑓(𝑧)=∑^∞_𝑛=1𝑛^𝑘(𝑛+1)(𝛼)_𝑛+1/𝛽_𝑛+1(𝜆/(𝑛+1+))^𝜇𝑎_𝑛𝑧^𝑛(𝐴−𝐵)+∑^∞_𝑛=1𝑛^𝑘(𝐴+𝑛𝐵)(𝜆/(𝑛+1+))^𝜇(𝛼)_𝑛+1/𝛽_𝑛+1𝑎_𝑛𝑧^𝑛≤1(𝑧∈𝑈). (3.4) Since |ℜ(𝑧)|≤|𝑧| for any 𝑧, therefore, we have ℜ∑^∞_𝑛=1𝑛^𝑘(𝑛+1)(𝛼)_𝑛+1/𝛽_𝑛+1(𝜆/(𝑛+1+𝜆))^𝜇𝑎_𝑛𝑧^𝑛(𝐴−𝐵)+∑^∞_𝑛=1𝑛^𝑘(𝐴+𝑛𝐵)(𝜆/(𝑛+1+𝜆))^𝜇(𝛼)_𝑛+1/𝛽_𝑛+1𝑎_𝑛𝑧^𝑛≤1(𝑧∈𝑈). (3.5) Choosing 𝑧 to be real and letting 𝑧→1 through real values, (3.5) yields ^∞_𝑛=1𝑛^𝑘()𝛼)𝑛+1(_𝑛+1𝛽_𝑛+1𝜆𝑛+1+𝜆^𝜇𝑎_𝑛≤(𝐴−𝐵)+^∞_𝑛=1𝑛^𝑘()𝜆𝐴+𝑛𝐵𝑛+1+𝜆^𝜇(𝛼)_𝑛+1𝛽_𝑛+1𝑎_𝑛, (3.6) which leads us to the desired inequality (3.2).
Conversely, by applying hypothesis (3.2), we get 𝑧𝐼^𝜇,𝑘_{𝛼,𝛽,𝜆}𝑓(𝑧)^′+𝐼^𝜇,𝑘_{𝛼,𝛽,𝜆}𝑓(𝑧)𝐵𝑧𝐼^𝜇,𝑘_{𝛼,𝛽,𝜆}𝑓(𝑧)^′+𝐴𝐼^𝜇,𝑘_{𝛼,𝛽,𝜆}𝑓(𝑧)≤∑^∞_𝑛=1𝑛^𝑘(𝑛+1)(𝛼)_𝑛+1/𝛽_𝑛+1(𝜆/(𝑛+1+𝜆))^𝜇𝑎_𝑛(𝐴−𝐵)+∑^∞_𝑛=1𝑛^𝑘(𝐴+𝑛𝐵)(𝛼)_𝑛+1/𝛽_𝑛+1(𝜆/(𝑛+1+𝜆))^𝜇𝑎_𝑛≤1(𝑧∈𝑈). (3.7) Hence, we have 𝑓(𝑧)∈Σ^𝜇,𝑘,+_{𝛼,𝛽,𝜆}(𝐴,𝐵). By observing that the function 𝑓(𝑧), given by (3.3), is indeed an extremal function for the assertion (3.2), we complete the proof of Theorem 3.1.

By applying Theorem 3.1, we obtain the following sharp coefficient estimates.

Corollary 3.2. Let 𝑓∈Σ be given by (1.13). If 𝑓∈Σ^𝜇,𝑘,+_{𝛼,𝛽,𝜆}(𝐴,𝐵), then 𝑎_𝑛≤(𝐴−𝐵)(𝑛+1+𝜆)^𝜇𝛽_𝑛+1𝑛^𝑘𝜆^𝜇[𝑛(1−𝐵)+(1−𝐴)](𝛼)_𝑛+1(𝑛≥1,𝑧∈𝑈), (3.8) where the equality holds true for the function 𝑓(𝑧) given by (3.3).

Next, we prove the following growth and distortion properties for the class Σ^𝜇,𝑘,+_{𝛼,𝛽,𝜆}.

Theorem 3.3. If the function 𝑓 defined by (1.13) is in the class Σ^𝜇,𝑘,+_{𝛼,𝛽,𝜆}(𝐴,𝐵), then, for 0<|𝑧|=𝑟<1, we have 1𝑟−(2+𝜆)^𝜇(𝐴−𝐵)𝛽₂𝜆^𝜇[2−(𝐴+𝐵)](𝛼)₂𝑓(𝑧≤1𝑟≤)𝑟+(2+𝜆)^𝜇(𝐴−𝐵)𝛽₂𝜆^𝜇[2−(𝐴+B)](𝛼)₂1𝑟,𝑟²−(2+𝜆)^𝜇(𝐴−𝐵)𝛽₂𝜆^𝜇[2−(𝐴+𝐵)](𝛼)₂≤(𝑧≤1𝑓′)𝑟²+(2+𝜆)^𝜇(𝐴−𝐵)𝛽₂𝜆^𝜇[2−(𝐴+𝐵)](𝛼)₂. (3.9) Each of these results is sharp with the extremal function 𝑓(𝑧) given by (3.3).

Proof. Since 𝑓∈Σ^𝜇,𝑘,+_{𝛼,𝛽,𝜆}(𝐴,𝐵). Theorem 3.1 readily yields the inequality ^∞_𝑛=1𝑎_𝑛≤(2+𝜆)^𝜇(𝐴−𝐵)𝛽₂𝜆^𝜇[2−(𝐴+𝐵)](𝛼)₂. (3.10) Thus, for 0<|𝑧|=𝑟<1 and utilizing (3.10), we have 𝑓(𝑧≤1)𝑧+∑^∞_𝑛=1𝑎_𝑛𝑧^𝑛≤1𝑟+𝑟∑^∞_𝑛=1𝑎_𝑛≤1𝑟()+𝑟2+𝜆^𝜇(𝐴−𝐵)𝛽₂𝜆^𝜇[2−(𝐴+𝐵)](𝛼)₂,𝑓(𝑧≥1)𝑧−^∞_𝑛=1𝑎_𝑛𝑧^𝑛≥1𝑟−𝑟^∞_𝑛=1𝑎_𝑛≥1𝑟()−𝑟2+𝜆^𝜇(𝐴−𝐵)𝛽₂𝜆^𝜇[2−(𝐴+𝐵)](𝛼)₂. (3.11)
Also from Theorem 3.1, we get ^∞_𝑛=1𝑛𝑎_𝑛≤(2+𝜆)^𝜇(𝐴−𝐵)𝛽₂𝜆^𝜇[2−(𝐴+𝐵)](𝛼)₂. (3.12) Hence (𝑧≤1𝑓′)𝑧²+∑^∞_𝑛=1𝑛𝑎_𝑛𝑧^𝑛−1≤1𝑟²+∑^∞_𝑛=1𝑛𝑎_𝑛≤1𝑟²+(2+𝜆)^𝜇(𝐴−𝐵)𝛽₂𝜆^𝜇[2−(𝐴+𝐵)](𝛼)₂,(𝑧≥1𝑓′)𝑧²−^∞_𝑛=1𝑛𝑎_𝑛𝑧^𝑛−1≥1𝑟²−^∞_𝑛=1𝑛𝑎_𝑛≥1𝑟²−(2+𝜆)^𝜇(𝐴−𝐵)𝛽₂𝜆^𝜇[2−(𝐴+𝐵)](𝛼)₂. (3.13) This completes the proof of Theorem 3.3.

We conclude this section by determining the radii of meromorphically univalent starlikeness and meromorphically univalent convexity of the class Σ^𝜇,𝑘,+_{𝛼,𝛽,𝜆}(𝐴,𝐵). We state our results as in the following theorems.

Theorem 3.4. Let 𝑓∈Σ^𝜇,𝑘,+_{𝛼,𝛽,𝜆}(𝐴,𝐵). Then, 𝑓 is meromorphically univalent starlike of order 𝛾(0≤𝛾<1) in |𝑧|<r₁, where 𝑟₁=𝑟₁𝐴,𝐵,𝛾=inf_𝑛≥01−𝛾[𝑛(1−𝐵)+(1−𝐴)]𝑛+2−𝛾(𝐴−𝐵)^𝑛+1. (3.14) The equality is attained for the function 𝑓(𝑧) given by (3.3).

Proof. It suffices to prove that 𝑧𝐼^𝜇,𝑘_{𝛼,𝛽,𝜆}^′𝐼^𝜇,𝑘_{𝛼,𝛽,𝜆}+1≤1−𝛾, (3.15) for |𝑧|<𝑟₁, we have 𝑧𝐼^𝜇,𝑘_{𝛼,𝛽,𝜆}^′𝐼^𝜇,𝑘_{𝛼,𝛽,𝜆}=∑+1^∞𝑛=1𝑛^𝑘(𝑛+1)(𝛼)_𝑛+1/𝛽_𝑛+1(𝜆/(𝑛+1+𝜆))^𝜇𝑎_𝑛𝑧^𝑛(1/𝑧)+∑^∞_𝑛=1𝑛^𝑘(𝛼)_𝑛+1/𝛽_𝑛+1(𝜆/(𝑛+1+𝜆))^𝜇𝑎_𝑛𝑧^𝑛≤∑^∞_𝑛=1𝑛^𝑘(𝑛+1)(𝛼)_𝑛+1/𝛽_𝑛+1(𝜆/(𝑛+1+𝜆))^𝜇𝑎_𝑛𝑧^𝑛+11−∑^∞_𝑛=1𝑛^𝑘(𝛼)_𝑛+1/𝛽_𝑛+1(𝜆/(𝑛+1+𝜆))^𝜇𝑎_𝑛𝑧^𝑛+1. (3.16) Hence, (3.16) holds true if ^∞_𝑛=1𝑛^𝑘()𝛼)𝑛+1(_𝑛+1𝛽_𝑛+1𝜆𝑛+1+𝜆^𝜇𝑎_𝑛𝑧^𝑛+1≤1−𝛾1−^∞_𝑛=1𝑛^𝑘(𝛼)_𝑛+1𝛽_𝑛+1𝜆𝑛+1+𝜆^𝜇𝑎_𝑛𝑧^𝑛+1 (3.17) or ^∞_𝑛=1𝑛^𝑘𝑛+2−𝛾𝛼)1−𝛾(_𝑛+1𝛽_𝑛+1𝜆𝑛+1+𝜆^𝜇𝑎_𝑛𝑧^𝑛+1≤1, (3.18) with the aid of (3.18) and (3.2), it is true to have 𝑛^𝑘𝑛+2−𝛾𝛼)1−𝛾(_𝑛+1𝛽_𝑛+1𝜆𝑛+1+𝜆^𝜇𝑧^𝑛+1≤𝑛^𝑘𝜆^𝜇[𝑛(1−𝐵)+(1−𝐴)](𝛼)_𝑛+1(𝐴−𝐵)(𝑛+1+𝜆)^𝜇𝛽_𝑛+1(𝑛≥1). (3.19) Solving (3.19) for |𝑧|, we obtain |𝑧|≤1−𝛾[𝑛(1−𝐵)+(1−𝐴)]𝑛+2−𝛾(𝐴−𝐵)^𝑛+1(𝑛≥1). (3.20) This completes the proof of Theorem 3.4.

Theorem 3.5. Let 𝑓∈Σ^𝜇,𝑘,+_{𝛼,𝛽,𝜆}(𝐴,𝐵). Then, 𝑓 is meromorphically univalent convex of order 𝛾(0≤𝛾<1) in |𝑧|<𝑟₂, where 𝑟₂=𝑟₂𝐴,𝐵,𝛾=inf_𝑛≥01−𝛾𝑛^𝑘−1(1−𝐵)+(1−𝐴)𝑛+2−𝛾(𝐴−𝐵)^𝑛+1. (3.21) The equality is attained for the function 𝑓(𝑧) given by (3.3).

Proof. By using the technique employed in the proof of Theorem 3.4, we can show that 𝑧𝐼^𝜇,𝑘,+_{𝛼,𝛽,𝜆}^′′𝐼^𝜇,𝑘,+_{𝛼,𝛽,𝜆}^′+2≤1−𝛾, (3.22) for |𝑧|<𝑟₂, with the aid of Theorem 3.1. Thus, we have the assertion of Theorem 3.5.

Acknowledgment

The work presented here was fully supported by UKM-ST-06-FRGS0244-2010.