A Subclass of Harmonic Univalent Functions Associated with 𝑞-Analogue of Dziok-Srivastava Operator

Abstract

We study a class of complex-valued harmonic univalent functions using a generalized operator involving basic hypergeometric function. Precisely, we give a necessary and sufficient coefficient condition for functions in this class. Distortion bounds, extreme points, and neighborhood of such functions are also considered.

1. Introduction

Let 𝕌={𝑧∈ℂ∶|𝑧|<1} be the open unit disc, and let 𝑆_𝐻 denote the class of functions which are complex valued, harmonic, univalent, and sense preserving in 𝕌 normalized by 𝑓(0)=𝑓_𝑧(0)−1=0. Each 𝑓∈𝑆_𝐻 can be expressed as 𝑓=ℎ+𝑔, where ℎ and 𝑔 are analytic in 𝕌. We call ℎ the analytic part and 𝑔 the coanalytic part of 𝑓. A necessary and sufficient condition for 𝑓 to be locally univalent and sense preserving in 𝕌 is that |ℎ^′(𝑧)|>|𝑔^′(𝑧)| in 𝕌 (see [1]). In [2], there is a more comprehensive study on harmonic univalent functions. Thus, for 𝑓=ℎ+𝑔∈𝑆_𝐻, we may write ℎ(𝑧)=𝑧+^∞_𝑘=2𝑎_𝑘𝑧^𝑘,𝑔(𝑧)=^∞_𝑘=1𝑏_𝑘𝑧^𝑘0≤𝑏₁<1. (1) Note that 𝑆_𝐻 reduces to 𝑆, the class of normalized analytic univalent functions, if the coanalytic part of 𝑓=ℎ+𝑔 is identically zero.

The study of basic hypergeometric series (also called 𝑞-hypergeometric series) essentially started in 1748 when Euler considered the infinite product (𝑞;𝑞)_∞⁻¹=∏^∞_𝑘=0(1−𝑞^𝑘+1)⁻¹. In the literature, we were told that the development of these functions was much slower until, in 1878, Heine converted a simple observation that lim_𝑞→1[(1−𝑞^𝑎)/(1−𝑞)]=𝑎 which returns the theory of₂𝜙₁ basic hypergeometric series to the famous theory of Gauss’s ₂𝐹₁ hypergeometric series. The importance of basic hypergeometric functions is due to their application in deriving 𝑞-analogue of well-known functions, such as 𝑞-analogues of the exponential, gamma, and beta functions. In this paper, we define a class of starlike harmonic functions using basic hypergeometric functions and investigate its properties like coefficient condition, distortion theorem, and extreme points.

For complex parameters 𝑎_𝑖, 𝑏_𝑗, 𝑞(𝑖=1,…,𝑟, 𝑗=1,…,𝑠, 𝑏_𝑗∈ℂ⧵{0,−1,−2,…}, |𝑞|<1), we define the basic hypergeometric function _𝑟Φ_𝑠(𝑎₁,…,𝑎_𝑟;𝑏₁,…,𝑏_𝑠,𝑞,𝑧) by _𝑟Φ_𝑠𝑎₁…,𝑎_𝑟;𝑏₁,…,𝑏_𝑠=,𝑞,𝑧_∞𝑘=0𝑎₁,𝑞_𝑘⋯𝑎_𝑟,𝑞_𝑘𝑞,𝑞_𝑘𝑏₁,𝑞_𝑘⋯𝑏_𝑠,𝑞_𝑘𝑧^𝑘, (2)(𝑟=𝑠+1;𝑟,𝑠∈ℕ₀=ℕ∪{0};𝑧∈𝕌) where ℕ denote the set of positive integers and (𝑎,𝑞)_𝑘 is the 𝑞-shifted factorial defined by 𝑎,𝑞_𝑘=()1,𝑘=0;1−𝑎1−𝑎𝑞1−𝑎𝑞²⋯1−𝑎𝑞^𝑘−1,𝑘∈ℕ. (3) We note that lim_{𝑞→1−𝑟}Φ_𝑠𝑞^𝑎1,…,𝑞^𝑎𝑟;𝑞^𝑏1,…,𝑞^𝑏𝑠,𝑞,𝑞−1^{1+𝑠−𝑟}𝑧=_𝑟𝐹_𝑠𝑎₁,…,𝑎_𝑟;𝑏₁,…,𝑏_𝑠,𝑧, (4) where _𝑟𝐹_𝑠(𝑎₁,…,𝑎_𝑟;𝑏₁,…,𝑏_𝑠,𝑧) is the well-known generalized hypergeometric function. By the ratio test, one observes that for |𝑞|<1 and 𝑟=𝑠+1 the series defined in (2) converges absolutely in 𝕌 so that it represented an analytic function in 𝕌. For more mathematical background of basic hypergeometric functions, one may refer to [3, 4].

The 𝑞-derivative of a function ℎ(𝑥) is defined by 𝐷_𝑞(ℎ(𝑥=ℎ(𝑥)))𝑞𝑥−ℎ𝑞−1𝑥,𝑞≠1,𝑥≠0. (5) For a function ℎ(𝑧)=𝑧^𝑘, we can observe that 𝐷_𝑞(ℎ(𝑧))=𝐷_𝑞𝑧^𝑘=1−𝑞^𝑘𝑧1−𝑞^𝑘−1=[𝑘]_𝑞𝑧^𝑘−1. (6) Then lim_𝑞→1𝐷_𝑞(ℎ(𝑧))=lim_𝑞→1[𝑘]_𝑞𝑧^𝑘−1=𝑘𝑧^𝑘−1=ℎ^′(𝑧), where ℎ^′(𝑧) is the ordinary derivative. For more properties of 𝐷_𝑞, see [4, 5].

Corresponding to the function _𝑟Φ_𝑠(𝑎₁,…,𝑎_𝑟;𝑏₁,…,𝑏_𝑠,𝑞,𝑧), consider _𝑟𝒢_𝑠𝑎₁,…,𝑎_𝑟;𝑏₁,…,𝑏_𝑠,𝑞,𝑧=𝑧_𝑟Φ_𝑠𝑎₁,…,𝑎_𝑟;𝑏₁,…,𝑏_𝑠,𝑞,𝑧=𝑧+^∞_𝑘=2𝑎₁,𝑞_𝑘−1⋯𝑎_𝑟,𝑞_𝑘−1𝑞,𝑞_𝑘−1𝑏₁,𝑞_𝑘−1⋯𝑏_𝑠,𝑞_𝑘−1𝑧^𝑘. (7) The authors [6] defined the linear operator 𝐻^𝑟_𝑠(𝑎₁,…,𝑎_𝑟;𝑏₁,…,𝑏_𝑠;𝑞)𝑓∶𝒜→𝒜 by 𝐻^𝑟_𝑠𝑎₁,…,𝑎_𝑟;𝑏₁,…,𝑏_𝑠𝑓(𝑧)=;𝑞_𝑟𝒢_𝑠𝑎₁,…,𝑎_𝑟;𝑏₁,…,𝑏_𝑠,𝑞,𝑧∗𝑓(𝑧)=𝑧+^∞_𝑘=2Γ𝑎₁,𝑞,𝑘𝑎_𝑘𝑧^𝑘, (8) where (∗) stands for convolution and Γ𝑎₁=𝑎,𝑞,𝑘¹,𝑞_𝑘−1⋯𝑎_𝑟,𝑞_𝑘−1𝑞,𝑞_𝑘−1𝑏₁,𝑞_𝑘−1⋯𝑏_𝑠,𝑞_𝑘−1. (9) To make the notation simple, we write 𝐻^𝑟_𝑠𝑎₁,𝑞𝑓(𝑧)=𝐻^𝑟_𝑠𝑎₁,…,𝑎_𝑟;𝑏₁,…,𝑏_𝑠;𝑞𝑓(𝑧). (10) We define the operator (8) of harmonic function 𝑓=ℎ+𝑔 given by (1) as 𝐻^𝑟_𝑠𝑎₁,𝑞𝑓(𝑧)=𝐻^𝑟_𝑠𝑎₁,𝑞ℎ(𝑧)+𝐻^𝑟_𝑠𝑎₁,𝑞𝑔(𝑧). (11)

Definition 1. For 0≤𝛿<1, let 𝑆^∗_𝐻(𝑎₁,𝛿,𝑞) denote the subfamily of starlike harmonic functions 𝑓∈𝑆^∗_𝐻 of the form (1) such that 𝜕𝜕𝜃arg𝐻^𝑟_𝑠𝑎₁,𝑞𝑓≥𝛿,|𝑧|=𝑟<1. (12) Following [7], a function 𝑓 is said to be in the class 𝑉_𝐻(𝑎₁,𝛿,𝑞)=𝑆^∗_𝐻(𝑎₁,𝛿,𝑞)∩𝑉_𝐻 if 𝑓 of the form (1) satisfies the condition that arg𝑎_𝑘=𝜃_𝑘arg𝑏_𝑘=𝜗_𝑘(𝑘≥𝑛+1;𝑛∈ℕ) (13) and if there exists a real number 𝜌 such that 𝜃_𝑘+(𝑘−1)𝜙≡𝜋(mod2𝜋),𝜗_𝑘+()().𝑘−1𝜙≡0,𝑘≥𝑛+1;𝑛∈ℕ (14) By specializing the parameters of 𝐻^𝑟_𝑠[𝑎₁,𝑞]𝑓, we obtain different classes of starlike harmonic functions, for example,(i)for 𝑟=𝑠+1,𝑎₂=𝑏₁,…,𝑎_𝑟=𝑏_𝑠, 𝑆^∗_𝐻(𝑞,𝑞,𝛿)=𝑆𝐻(𝛿) [8] is the class of sense-preserving harmonic univalent functions 𝑓 which are starlike of order 𝛿 in 𝕌; that is, 𝜕/𝜕𝜃(arg𝑓(𝑟𝑒^𝑖𝜃))≥𝛿;(ii)for 𝑟=𝑠+1, 𝑎₂=𝑏₁,…,𝑎_𝑟=𝑏_𝑠, and 𝑎₁=𝑞^𝑛+1,𝑞→1,  𝑆^∗_𝐻(𝑞^𝑛+1,𝑞,𝛿)=𝑅_𝐻(𝑛,𝛼) [9] is the class of starlike harmonic univalent functions with 𝜕/𝜕𝜃(arg𝐷^𝑛𝑓(𝑧))≥𝛿, where 𝐷 is the Ruscheweyh derivative (see [10]); (iii)for 𝑖={1,…,𝑟},𝑗={1,…,𝑠},𝑟=𝑠+1,𝑎_𝑖=𝑞^𝛼𝑖, and 𝑏_𝑗=𝑞^𝛽𝑗,𝑞→1,𝑆^∗_𝐻(𝑎₁,𝑞,𝛿)=𝑆^∗_𝐻(𝛼₁,𝛿) [11] is the class of starlike harmonic univalent functions with (𝜕/𝜕𝜃)(arg𝐻^𝑟_𝑠[𝛼₁]𝑓)≥𝛿, where 𝐻^𝑟_𝑠[𝛼₁] is the Dziok-Srivastava operator (see [12]).

2. Main Results

In our first theorem, we introduce a sufficient coefficient bound for harmonic functions in 𝑆^∗_𝐻(𝑎₁,𝛿,𝑞).

Theorem 2. Let 𝑓=ℎ+𝑔 be given by (1). If ^∞_𝑘=2[𝑘]_𝑞−𝛿𝑎1−𝛿^𝑘+[𝑘]_𝑞+𝛿𝑏1−𝛿^𝑘Γ𝑎₁,𝑞,𝑘≤1−1+𝛿𝑏1−𝛿¹, (15) where 𝑎₁=1,0≤𝛿<1, and Γ(𝑎₁,𝑞,𝑘) is given by (9), then 𝑓∈𝑆^∗_𝐻(𝑎₁,𝛿,𝑞).

Proof. To prove that 𝑓∈𝑆^∗_𝐻(𝑎₁,𝛿,𝑞), we only need to show that if (15) holds, then the required condition (12) is satisfied. For (12), we can write 𝜕𝜕𝜃arg𝐻^𝑟_𝑠𝑎₁,𝑞𝑓(𝑧)=ℜ𝑧𝐷_𝑞𝐻^𝑟_𝑠𝑎₁,𝑞ℎ(𝑧)𝐻^𝑟_𝑠𝑎₁,𝑞ℎ(𝑧)+𝐻^𝑟_𝑠𝑎₁𝑔(𝑧)−,𝑞𝑧𝐷_𝑞𝐻^𝑟_𝑠𝑎₁,𝑞𝑔(𝑧)𝐻^𝑟_𝑠𝑎₁,𝑞ℎ(𝑧)+𝐻^𝑟_𝑠𝑎₁𝑔(𝑧)𝐴(𝑧),𝑞=ℜ𝐵(𝑧). (16) Using the fact that ℜ(𝑤)≥𝛿 if and only if |1−𝛿+𝑤|≥|1+𝛿−𝑤|, it suffices to show that |𝐴(𝑧)+(1−𝛿)𝐵(𝑧)|−|𝐴(𝑧)−(1+𝛿)𝐵(𝑧)|≥0. (17) Substituting for 𝐴(𝑧) and 𝐵(𝑧) in (15) yields |𝐴(𝑧)+()𝐵(𝑧−|𝐴(𝑧)−()𝐵(𝑧≥(𝑧|−1−𝛿)|1+𝛿)|2−𝛿)|^∞_𝑘=2[𝑘]_𝑞+1−𝛿Γ𝑎₁,𝑞,𝑘𝑎_𝑘|𝑧|^𝑘−^∞_𝑘=1[𝑘]_𝑞−1+𝛿Γ𝑎₁,𝑞,𝑘𝑏_𝑘|𝑧|^𝑘−𝛿|𝑧|−^∞_𝑘=2[𝑘]_𝑞−1−𝛿Γ𝑎₁,𝑞,𝑘𝑎_𝑘|𝑧|^𝑘−^∞_𝑘=1[𝑘]_𝑞+1+𝛿Γ𝑎₁,𝑞,𝑘𝑏_𝑘|𝑧|^𝑘≥2(1−𝛿)|𝑧|1−^∞_𝑘=2[𝑘]_𝑞−𝛿Γ𝑎1−𝛿¹,𝑞,𝑘𝑎_𝑘−^∞_𝑘=1[𝑘]_𝑞+𝛿Γ𝑎1−𝛿¹,𝑞,𝑘𝑏_𝑘=2(1−𝛿)|𝑧|1−1+𝛿𝑏1−𝛿¹−^∞_𝑘=2[𝑘]_𝑞−𝛿𝑎1−𝛿^𝑘+[𝑘]_𝑞+𝛿𝑏1−𝛿^𝑘×Γ𝑎₁,𝑞,𝑘. (18) The last expression is nonnegative by (15), and so, 𝑓∈𝑆^∗_𝐻(𝑎₁,𝛿,𝑞).

Now, we obtain the necessary and sufficient conditions for 𝑓=ℎ+g given by (14).

Theorem 3. Let 𝑓=ℎ+𝑔 be given by (11). Then, 𝑓∈𝑉_𝐻(𝑎₁,𝛿,𝑞) if and only if ^∞_𝑘=2[𝑘]_𝑞−𝛿𝑎1−𝛿^𝑘+[𝑘]_𝑞+𝛿𝑏1−𝛿^𝑘Γ𝑎₁,𝑞,𝑘≤1−1+𝛿𝑏1−𝛿¹, (19) where 𝑎₁=1,0≤𝛿<1, and Γ(𝑎₁,𝑞,𝑘) is given by (9).

Proof. Since 𝑉_𝐻(𝑎₁,𝛿,𝑞)⊂𝑆^∗_𝐻(𝑎₁,𝛿,𝑞), we only to prove the only if part of the theorem. So that for functions 𝑓∈𝑉_𝐻(𝑎₁,𝛿,𝑞), we notice that the condition (𝜕/𝜕𝜃)(arg𝐻^𝑟_𝑠[𝑎₁,𝑞]𝑓(𝑧))≥𝛿 is equivalent to 𝜕𝜕𝜃arg𝐻^𝑟_𝑠𝑎₁,𝑞𝑓(𝑧)−𝛿=ℜ𝑧𝐷_𝑞𝐻^𝑟_𝑠𝑎₁,𝑞ℎ(𝑧)𝐻^𝑟_𝑠𝑎₁,𝑞ℎ(𝑧)+𝐻^𝑟_𝑠𝑎₁𝑔(𝑧)−,𝑞𝑧𝐷_𝑞𝐻^𝑟_𝑠𝑎₁,𝑞𝑔(𝑧)𝐻^𝑟_𝑠𝑎₁,𝑞ℎ(𝑧)+𝐻^𝑟_𝑠𝑎₁,𝑞𝑔(𝑧)−𝛿≥0. (20) That is, ℜ(1−𝛿)𝑧+∑^∞_𝑘=2[𝑘]_𝑞−𝛿Γ𝑎₁,𝑞,𝑘𝑎_𝑘𝑧^𝑘−∑^∞_𝑘=1[𝑘]_𝑞+𝛿Γ𝑎₁,𝑞,𝑘𝑏_𝑘𝑧^𝑘𝑧+∑^∞_𝑘=2Γ𝑎₁,𝑞,𝑘𝑎_𝑘𝑧^𝑘+∑^∞_𝑘=1Γ𝑎₁,𝑞,𝑘𝑏_𝑘𝑧^𝑘≥0. (21)
The previous condition must hold for all values of 𝑧 in 𝕌. Upon choosing 𝜙 according to (14), we must have (1−𝛿)−(1+𝛿)𝑏₁1+𝑏₁+∑^∞_𝑘=2𝑎_𝑘+𝑏_𝑘Γ𝑎₁,𝑞,𝑘𝑟^𝑘−1−∑^∞_𝑘=2[𝑘]_𝑞−𝛿𝑎_𝑘+[𝑘]_𝑞+𝛿𝑏_𝑘Γ𝑎₁,𝑞,𝑘𝑟^𝑘−11+𝑏₁+∑^∞_𝑘=2𝑎_𝑘+𝑏_𝑘Γ𝑎₁,𝑞,𝑘𝑟^𝑘−1≥0. (22) If condition (19) does not hold, then the numerator in (22) is negative for 𝑟 sufficiently close to 1. Hence, there exist 𝑧₀=𝑟₀ in (0,1) for which the quotient of (22) is negative. This contradicts the fact that 𝑓∈𝑉_𝐻(𝑎₁,𝛿,𝑞), and this completes the proof.

The following theorem gives the distortion bounds for functions in 𝑉_𝐻(𝑎₁,𝛿,𝑞) which yield a covering result for this class.

Theorem 4. If 𝑓∈𝑉_𝐻(𝑎₁,𝛿,𝑞), then 𝑓(𝑧)≤1+𝑏₁𝑟+1Γ𝑎₁,𝑞,21−𝛿−𝑞+1−𝛿1+𝛿𝑟𝑞+1−𝛿²|𝑧|𝑓(𝑧≥𝑏=𝑟<1,)1+₁𝑟−1Γ𝑎₁,𝑞,21−𝛿−𝑞+1−𝛿1+𝛿𝑟𝑞+1−𝛿²|𝑧|=𝑟<1, (23) where Γ𝑎₁,𝑞,2=1−𝑎₁⋯1−𝑎_𝑟1−𝑞1−𝑏₁⋯1−𝑏_𝑠,[2]_𝑞=𝑞+1. (24)

Proof. We will only prove the right hand inequality. The proof for the left hand inequality is similar.
let 𝑓∈𝑉_𝐻(𝑎₁,𝛿,𝑞). Taking the absolute value of 𝑓, we obtain 𝑓(𝑧)≤1+𝑏₁𝑟+^∞_𝑘=2𝑎_𝑘+𝑏_𝑘𝑟^𝑘≤1+𝑏₁𝑟+^∞_𝑘=2𝑎_𝑘+𝑏_𝑘𝑟². (25) That is, 𝑓(𝑧)≤1+𝑏₁𝑟+1−𝛿Γ𝑎₁,𝑞,2[2]_𝑞×−𝛿_∞𝑘=2[2]_𝑞−𝛿𝑎1−𝛿^𝑘+[2]_𝑞−𝛿𝑏1−𝛿^𝑘×Γ𝑎₁,𝑞,2𝑟²≤1+𝑏₁𝑟+1−𝛿Γ𝑎₁×,𝑞,2𝑞+1−𝛿1−1+𝛿𝑏1−𝛿¹𝑟²=1+𝑏₁1𝑟+Γ𝑎₁×,𝑞,21−𝛿−𝑞+1−𝛿1+𝛿𝑏𝑞+1−𝛿¹𝑟². (26)

Corollary 5. Let 𝑓 be of the form (1) so that 𝑓∈𝑉_𝐻(𝑎₁,𝛿,𝑞). Then, |𝑤|<𝑎𝑤∶2Γ₁,𝑞,2−1−Γ𝑎₁,𝑞,2−1𝛿𝑞+1−𝛿Γ𝑎₁−𝑎,𝑞,22Γ₁,𝑞,2−1−Γ𝑎₁,𝑞,2−1𝛿𝑞+1+𝛿Γ𝑎₁𝑏,𝑞,2¹⊂𝑓(𝕌). (27) Next, one determines the extreme points of closed convex hull of 𝑉_𝐻(𝑎₁,𝛿,𝑞) denoted by clco𝑉_𝐻(𝑎₁,𝛿,𝑞).

Theorem 6. Set 𝜆_𝑘=1−𝛿[𝑘]_𝑞−𝛿Γ𝑎₁,𝜇,𝑞,𝑘_𝑘=1−𝛿[𝑘]_𝑞+𝛿Γ𝑎₁.,𝑞,𝑘 (28) For 𝑏₁ fixed, the extreme points for clco𝑉_𝐻(𝑎₁,𝛿,𝑞) are 𝑧+𝜆_𝑘𝑥𝑧^𝑘+𝑏₁𝑧∪𝑧+𝑏₁𝑧+𝜇_𝑘𝑥𝑧^𝑘, (29) where 𝑘≥2 and |𝑥|=1−|𝑏₁|.

Proof. Any function 𝑓∈clco𝑉_𝐻(𝑎₁,𝛿,𝑞) may be expressed as 𝑓(𝑧)=𝑧+^∞_𝑘=2𝑎_𝑘𝑒^𝑖𝛼𝑘𝑧^𝑘+𝑏₁𝑧+^∞_𝑘=2𝑏_𝑘𝑒^𝑖𝛽𝑘𝑧^𝑘, (30) where the coefficients satisfy the inequality (15). Set ℎ₁(𝑧)=𝑧, 𝑔₁(𝑧)=𝑏₁𝑧,ℎ_𝑘(𝑧)=𝑧+𝜆_𝑘𝑒^𝑖𝛼𝑘𝑧^𝑘, and𝑔_𝑘(𝑧)=𝑏₁(𝑧)+𝑒^𝑖𝛽𝑘𝑧^𝑘, for 𝑘=2,3,…. Writing 𝑋_𝑘=|𝑎_𝑘|/𝜆_𝑘,𝑌_𝑘=|𝑏_𝑘|/𝜇_𝑘, 𝑘=2,3,… and 𝑋₁=1−∑^∞_𝑘=2𝑋_𝑘;𝑌₁=1−∑^∞_𝑘=2𝑌_𝑘,  we have 𝑓(𝑧)=^∞_𝑘=1𝑋_𝑘ℎ_𝑘(𝑧)+𝑌_𝑘𝑔_𝑘(𝑧). (31) In particular, set 𝑓₁(𝑧)=𝑧+𝑏₁𝑧,𝑓_𝑘(𝑧)=𝑧+𝜆_𝑘𝑥𝑧^𝑘+𝑏₁𝑧+𝜇_𝑘𝑦𝑧^𝑘,𝑘≥2,|𝑥|+𝑦=1−𝑏₁. (32)
Therefore, the extreme points of clco𝑉_𝐻(𝑎₁,𝛿,𝑞) are contained in {𝑓_𝑘(𝑧)}. To see that 𝑓₁ is not an extreme point, note that 𝑓₁ may be written as a convex linear combination of functions in clco𝑉_𝐻(𝑎₁,𝛿,𝑞) as follows: 𝑓₁(𝑧)=12𝑓₁(𝑧)+𝜆₂1−𝑏₁𝑧²+12𝑓₁(𝑧)−𝜆₂1−𝑏₁𝑧². (33) If both |𝑥|≠0 and |𝑦|≠0, we will show that it can also be expressed as a convex linear combination of functions in clco𝑉_𝐻(𝑎₁,𝛿,𝑞). Without loss of generality, assume that |𝑥|≥|𝑦|. Choose 𝜖>0 small enough so that 𝜖<|𝑥|/|𝑦|. Set 𝐴=1+𝜖 and 𝐵=1−|𝜖𝑥/𝑦|. We then see that both 𝑡₁(𝑧)=𝑧+𝜆_𝑘𝐴𝑥𝑧^𝑘+𝑏₁𝑧+𝜇_𝑘𝑦𝐵𝑧^𝑘,𝑡₂(𝑧)=𝑧+𝜆_𝑘(2−𝐴)𝑥𝑧^𝑘+𝑏₁𝑧+𝜇_𝑘𝑦(2−𝐵)𝑧^𝑘, (34) are in clco𝑉_𝐻(𝑎₁,𝛿,𝑞) and note that 𝑓_𝑛(𝑧)=12𝑡₁(𝑧)+𝑡₂(𝑧). (35) The extremal coefficient bound shows that the functions of the form (29) are extreme points for clco𝑉_𝐻(𝑎₁,𝛿,𝑞), and so the proof is complete.

Following the earlier works in [2, 13], we refer to the 𝛾-neighborhood of the functions 𝑓 defined by (1) to be the set of functions 𝐹 for which 𝑁_𝛾𝑓=𝐹=𝑧+^∞_𝑘=2𝐴_𝑘𝑧^𝑘+^∞_𝑘=1𝐵_𝑘𝑧^𝑘∶^∞_𝑘=2𝑘𝑎_𝑘−𝐴_𝑘+𝑏_𝑘−𝐵_𝑘+𝑏₁−𝐵₁≤𝛾. (36)

We define the 𝑞-𝛾-neighborhood of a function 𝑓∈𝑆_𝐻 as follows: 𝑁^𝑞_𝛾𝑓=𝐹=𝑧+^∞_𝑘=2𝐴_𝑘𝑧^𝑘+^∞_𝑘=1𝐵_𝑘𝑧^𝑘∶^∞_𝑘=2[𝑘]_𝑞𝑎_𝑘−𝐴_𝑘+𝑏_𝑘−𝐵_𝑘+𝑏₁−𝐵₁≤𝛾. (37)

In our case, let us define the generalized 𝑞-𝛾-neighborhood of 𝑓 to be the set 𝑁^𝑞_𝛾𝑓=𝐹∶^∞_𝑘=2Γ𝑎₁,𝑞,𝑘[𝑘]_𝑞−𝛿𝑎_𝑘−𝐴_𝑘+[𝑘]_𝑞+𝛿𝑏_𝑘−𝐵_𝑘+(1+𝛿)𝑏₁−𝐵₁≤(1−𝛿)𝛾. (38)

Theorem 7. Let 𝑓 be given by (1). If 𝑓 satisfies the conditions ^∞_𝑘=2[𝑘]_𝑞[𝑘]_𝑞−𝛿𝑎_𝑘Γ𝑎₁+,𝑞,𝑘_∞1=2[𝑘]_𝑞[𝑘]_𝑞+𝛿𝑏_𝑘Γ𝑎₁,𝑞,𝑘≤1−𝛿,0≤𝛿<1,𝛾≤1−𝛿𝑞+1−𝛿1−1+𝛿𝑏1−𝛿¹, (39) then 𝑁^𝑞_𝛾(𝑓)⊂𝑆^∗_𝐻(𝑎₁,𝛿,𝑞).

Proof. Let 𝑓 satisfy (39) and let 𝐹(𝑧)=𝑧+𝐵₁𝑧+^∞_𝑘=2𝐴_𝑘𝑧^𝑘+𝐵_𝑘𝑧^𝑘 (40) belong to 𝑁^𝑞_𝛾(𝑓). We have (1+𝛿)𝐵₁+^∞_𝑘=2Γ𝑎₁,𝑞,𝑘[𝑘]_𝑞−𝛿𝐴_𝑘+[𝑘]_𝑞+𝛿𝐵_𝑘≤(1+𝛿)𝐵₁−𝑏₁+(1+𝛿)𝑏₁+^∞_𝑘=2Γ𝑎₁,𝑞,𝑘[𝑘]_𝑞−𝛿𝐴_𝑘−𝑎_𝑘+[𝑘]_𝑞+𝛿𝐵_𝑘−𝑏_𝑘+^∞_𝑘=2Γ𝑎₁,𝑞,𝑘[𝑘]_𝑞−𝛿𝑎_𝑘+[𝑘]_𝑞+𝛿𝑏_𝑘≤(1−𝛿)𝛾+(1+𝛿)𝑏₁+1[2]_𝑞−𝛿^∞_𝑘=2Γ𝑎₁,𝑞,𝑘[𝑘]_𝑞[𝑘]_𝑞−𝛿𝑎_𝑘+[𝑘]_𝑞[𝑘]_𝑞+𝛿𝑏_𝑘≤(1−𝛿)𝛾+(1+𝛿)𝑏₁+1×)−(𝑏𝑞+1−𝛿(1−𝛿1+𝛿)₁≤1−𝛿. (41) Hence, 𝛾≤1−𝛿𝑞+1−𝛿1−1+𝛿𝑏1−𝛿¹,𝐹∈𝑆^∗_𝐻𝑎₁,𝛿,𝑞. (42)

Acknowledgment

The work presented here was partially supported by LRGS/TD/2011/UKM/ICT/03/02 and UKM-DLP-2011-050.