Fekete-Szegő Inequality for a Subclass of 𝑝-Valent Analytic Functions

Abstract

The main object of this paper is to study Fekete-Szegő problem for the class of 𝑝-valent functions. Fekete-Szegő inequality of several classes is obtained as special cases from our results. Applications of the results are also obtained on the class defined by convolution.

1. Introduction and Preliminaries

Let 𝐴_𝑝 denote the class of functions 𝑓(𝑧) of the form 𝑓(𝑧)=𝑧^𝑝+^∞_𝑛=𝑝+1𝑎_𝑛𝑧^𝑛,𝑝∈ℕ={1,2,3,…}, (1) which are analytic in the open unit disk 𝐸. Also, 𝐴₁=𝐴, the usual class of analytic functions defined in the open unit disk 𝐸={𝑧∶|𝑧|<1}. Let 𝑓(𝑧) and 𝑔(𝑧) be analytic in 𝐸. We say that the function 𝑓 is subordinate to the function 𝑔 and write 𝑓(𝑧)≺𝑔(𝑧), if and only if there exists Schwarz function 𝑤, analytic in 𝐸 such that 𝑤(0)=0, |𝑤(𝑧)|<1 for 𝑧∈𝐸, and 𝑓(𝑧)=𝑔(𝑤(𝑧)). In particular, if 𝑔 is univalent in 𝐸, then we have the following equivalence: 𝑓(𝑧)≺𝑔(𝑧)⟺𝑓(0)=𝑔(0),𝑓(𝐸)⊂𝑔(𝐸). (2)

For any two analytic functions 𝑓(𝑧) of the form (1) and 𝑔(𝑧) with 𝑔(𝑧)=𝑧^𝑝+^∞_𝑛=𝑝+1𝑏_𝑛𝑧^𝑛,𝑧∈𝐸, (3) the convolution (Hadamard product) is given by 𝑓∗𝑔(𝑧)=𝑧^𝑝+^∞_𝑛=𝑝+1𝑎_𝑛𝑏_𝑛𝑧^𝑛,𝑧∈𝐸. (4) Let 𝜙(𝑧) be an analytic function with positive real part on 𝐸 with 𝜙(0)=1, 𝜙^′(0)>0, which maps the unit disk 𝐸 onto a region starlike with respect to 1 which is symmetric with respect to the real axis. Denote by 𝑆^∗_𝑝(𝜙) the class of functions 𝑓 analytic in 𝐸 for which 𝑧𝑓^′(𝑧)(𝑧)(𝑧)𝑝𝑓≺𝜙,𝑧∈𝐸. (5) The class𝑆^∗_𝑝(𝜙) was defined and studied by Ali et al. [1]. They obtained the Fekete-Szegő inequality for functions in the class 𝑆^∗_𝑝(𝜙). The class 𝑆^∗₁(𝜙) coincides with the class 𝑆^∗(𝜙) discussed by Ma and Minda [2]. Owa [3] introduced a subclass of 𝑝-valently Bazilevic functions 𝐻_𝑝(𝐴,𝐵,𝛼,𝛽). A function 𝑓∈𝐴_𝑝 is said to be in the class 𝐻_𝑝(𝐴,𝐵,𝛼,𝛽) if and only if 1−𝛽𝑓(𝑧)𝑧^𝑝𝛼+𝛽𝑧𝑓^′(𝑧)(𝑧)𝑓(𝑧)𝑝𝑓𝑧^𝑝𝛼≺1+𝐴𝑧1+𝐵𝑧,𝑧∈𝐸, (6) where −1≤𝐵<𝐴≤1, 𝛼≥0, and 0≤𝛽≤1. We now define the following subclass of analytic functions.

Definition 1. Let 𝜙(𝑧) be a univalent starlike function with respect to 1 which maps the unit disk 𝐸 onto a region in the right half-plane which is symmetric with respect to the real axis with 𝜙(0)=1 and 𝜙^′(0)>0. A function 𝑓∈𝐴_𝑝 is in the class 𝑉_{𝑝,𝑏,𝛼,𝛽}(𝜙) if 21−𝑏+2𝑏𝑓(𝑧)1−𝛽𝑧^𝑝𝛼+𝛽𝑧𝑓^′(𝑧)(𝑧)𝑓(𝑧)𝑝𝑓𝑧^𝑝𝛼≺𝜙(𝑧), (7) where 0≤𝛽≤1, 𝛼≥0, and 𝑏>0.

Definition 2. A function 𝑓∈𝐴_𝑝 is in the class 𝑉_{𝑝,𝑏,𝛼,𝛽,𝑔}(𝜙) if 21−𝑏+2𝑏(𝑧)1−𝛽𝑓∗𝑔𝑧^𝑝𝛼𝑧+𝛽𝑓∗𝑔^′(𝑧)𝑝(𝑧)(𝑧)𝑓∗𝑔𝑓∗𝑔𝑧^𝑝𝛼≺𝜙(𝑧), (8) where 0≤𝛽≤1, 𝛼≥0, and 𝑏>0.In other words, a function 𝑓∈𝐴_𝑝is in the class 𝑉_{𝑝,𝑏,𝛼,𝛽,𝑔}(𝜙) if (𝑓∗𝑔)(𝑧)∈𝑉_{𝑝,𝑏,𝛼,𝛽}(𝜙).
We have the following special cases.(i)𝑉_𝑝,2,1,1(𝜙) coincides with the class 𝑆^∗_𝑝(𝜙) introduced and studied by Ali et al. [1].(ii)For 𝑝=1, 𝑏=2, and 𝛽=1, we have the class 𝐵_𝛼(𝜙) introduced and studied by Ravichandran et al. [4].(iii)For 𝑏=2 and 𝜙(𝑧)=(1+𝐴𝑧)/(1+𝐵𝑧), the class 𝑉_{𝑝,2,𝛼,𝛽}(𝜙) reduces to 𝐻_𝑝(𝐴,𝐵,𝛼,𝛽) introduced and studied by Owa [3].(iv)For 𝜙(𝑧)=(1+(1−2𝛾)𝑧)/(1−𝑧), the class 𝑉_{𝑝,2,𝛼,1}(𝜙) reduces to the class 𝐵_𝑝(𝛼,𝛾)defined as 𝐵_𝑝𝛼,𝛾=𝑓∈𝐴_𝑝∶Re𝑧𝑓^′(𝑧)(𝑧)𝑓(𝑧)𝑝𝑓𝑧^𝑝𝛼.>𝛾,0≤𝛾<1,𝑧∈𝐸 (9)(v)For 𝜙(𝑧)=(1+(1−2𝛾)𝑧)/(1−𝑧), the class 𝑉_{𝑝,2,𝛼,0}(𝜙) is defined as 𝑓∈𝐴_𝑝𝑓(𝑧)∶Re𝑧^𝑝𝛼>𝛾,0≤𝛾<1,𝑧∈𝐸. (10)(vi)𝑉_1,2,0,1(𝜙)=𝑆^∗(𝜙) is investigated by Ma and Minda [2].(vii)For 𝜙(𝑧)=(1+(1−2𝛾)𝑧)/(1−𝑧), the class 𝑉_1,2,1,0(𝜙) reduces to the class 𝐵_𝛾=𝑓∈𝐴₁𝑓(𝑧)∶Re𝑧>𝛾. (11)studied by Chen [5].

We need the following results to obtain our main results.

Lemma 3 (see [1]). Let Ω be the class of analytic functions 𝑤, normalized by 𝑤(0)=0,satisfying condition |𝑤(𝑧)|<1.If 𝑤∈Ωand 𝑤(𝑧)=𝑤₁𝑧+𝑤₂𝑧²+⋯, 𝑧∈𝐸,then 𝑤₂−𝑡𝑤²₁≤−𝑡,𝑡<−1,1,−1≤𝑡≤1,𝑡,𝑡>1. (12) For 𝑡≤−1 or 𝑡≥1,the equality holds, if and only if 𝑤(𝑧)=𝑧 or one of its rotation. For −1≤𝑡≤1,the equality holds, if 𝑤(𝑧)=𝑧² or one of its rotation. the equality holds for 𝑡=−1, if and only if 𝑤(𝑧)=𝑧((𝜆+𝑧)/(1+𝜆𝑧))(0≤𝜆≤1) or one of its rotation, while for 𝑡=1, the equality holds, if and only if 𝑤(𝑧)=−𝑧((𝜆+𝑧)/(1+𝜆𝑧))(0≤𝜆≤1) or one of its rotation. The above upper bound for −1<𝑡<1 is sharp, and it can be improved as follows: 𝑤₂−𝑡𝑤²₁+(1+𝑡)𝑤₁²𝑤≤1,−1<𝑡≤0,₂−𝑡𝑤²₁+(1−𝑡)𝑤₁²≤1,0<𝑡<1. (13)

Lemma 4 (see [6, (7), page 10]). If 𝑤∈Ωand 𝑤(𝑧)=𝑤₁𝑧+𝑤₂𝑧²+⋯, 𝑧∈𝐸,then 𝑤₂−𝑡𝑤²₁≤max{1;|𝑡|}, (14) for any complex number 𝑡. The result is sharp for the functions 𝑤(𝑧)=𝑧² or 𝑤(𝑧)=𝑧.

Lemma 5 (see [7]). If 𝑤∈Ω, then for any real number 𝑞₁ and 𝑞₂, the following sharp estimate holds: 𝑤₃+𝑞₁𝑤₁𝑤₂+𝑞₂𝑤³₁≤𝐻𝑞₁,𝑞₂, (15) where 𝐻𝑞₁,𝑞₂=𝑞1,₁,𝑞₂∈𝐷₁∪𝐷₂,𝑞₂,𝑞₁,𝑞₂∈⋃⁷_𝑘=3𝐷_𝑘,23𝑞₁𝑞+1¹+13𝑞₁+1+𝑞₂^1/2,𝑞₁,𝑞₂∈𝐷₈∪𝐷₉,13𝑞₂𝑞²₁−4𝑞²₁−4𝑞₂𝑞²₁−43𝑞₂−1^1/2,𝑞₁,𝑞₂∈𝐷₁₀∪𝐷₁₁⧵{},2±2,13𝑞₁𝑞−1¹−13𝑞₁−1−𝑞₂^1/2,𝑞₁,𝑞₂∈𝐷₁₂. (16) The extremal function up to the rotations is of the form 𝑤(𝑧)=𝑧³(𝑧)𝑤(𝑧),𝑤=𝑧,=𝑤₀(𝑧)=𝑧(1−𝜆)𝜀₂+𝜆𝜀₁−𝜀₁𝜀₂𝑧1−(1−𝜆)𝜀₁+𝜆𝜀₂𝑧,𝑤(𝑧)=𝑤₁(𝑧)=𝑧𝑡₁−𝑧1−𝑡₁𝑧,𝑤(𝑧)=𝑤₂(𝑧)=𝑧𝑡₂+𝑧1+𝑡₂𝑧,𝜀₁=𝜀₂=1,𝜀₁=𝑡₀−𝑒^{−𝑖𝜃0/2}(),𝜀𝑎∓𝑏₂=−𝑒^{−𝑖𝜃0/2}(𝑖𝑎±𝑏),𝑎=𝑡₀𝜃cos⁰2,𝑏=1−𝑡²₀sin²𝜃₀2,𝜆=𝑏±𝑎,𝑡2𝑏₀=2𝑞₂𝑞²₁+2−3𝑞²₁3𝑞₂−1𝑞²₁−4𝑞₂^1/2,𝑡₁=𝑞₁+13𝑞₁+1+𝑞₂^1/2,𝑡₂=𝑞₁−13𝑞₁−1−𝑞₂^1/2,𝜃cos⁰2=𝑞₁2𝑞₂𝑞²₁+8−2𝑞²₁+22𝑞₂𝑞²₁+2−3𝑞²₁. (17) The sets 𝐷_𝑘, 𝑘=1,2,…,12 are defined as follows: 𝐷₁=𝑞₁,𝑞₂∶𝑞₁≤12,𝑞₂,𝐷≤1₂=𝑞₁,𝑞₂∶12≤𝑞₁4≤2,𝑞27¹+1³−𝑞₁+1≤𝑞₂,𝐷≤1₃=𝑞₁,𝑞₂∶𝑞₁≤12,𝑞₂,𝐷≤−1₄=𝑞₁,𝑞₂∶𝑞₁≥12,𝑞₂2≤−3𝑞₁,𝐷+1₅=𝑞₁,𝑞₂∶𝑞₁≤2,𝑞₂,𝐷≥1₆=𝑞₁,𝑞₂∶2≤𝑞₁≤4,𝑞₂≥1𝑞12²¹,𝐷+8₇=𝑞₁,𝑞₂∶𝑞₁≥4,𝑞₂≥23𝑞₁,𝐷−1₈=𝑞₁,𝑞₂∶12≤𝑞₁−2≤2,3𝑞₁+1≤𝑞₂≤4𝑞27¹+1³−𝑞₁,𝐷+1₉=𝑞₁,𝑞₂∶𝑞₁−2≥2,3𝑞₁+1≤𝑞₂≤2𝑞₁𝑞₁+1𝑞²₁+2𝑞₁,𝐷+4₁₀=𝑞₁,𝑞₂∶2≤𝑞₁2𝑞≤4,₁𝑞₁+1𝑞²₁+2𝑞₁+4≤𝑞₂≤1𝑞12²¹,𝐷+8₁₁=𝑞₁,𝑞₂∶𝑞₁2𝑞≥4,₁𝑞₁+1𝑞²₁+2𝑞₁+4≤𝑞₂≤2𝑞₁𝑞₁−1𝑞²₁−2𝑞₁,𝐷+4₁₂=𝑞₁,𝑞₂∶𝑞₁2𝑞≥4,₁𝑞₁−1𝑞²₁−2𝑞₁+4≤𝑞₂≤23𝑞₁−1. (18)

2. Main Results

Theorem 6. Let 𝜙(𝑧)=1+𝐵₁𝑧+𝐵₂𝑧²+𝐵₃𝑧³+⋯, where 𝐵_𝑛, 𝑠 are real with 𝐵₁>0 and 𝐵₂≥0. Let 𝜎₁=𝛼𝑝+𝛽²𝑏𝑝𝐵²₁×2𝐵𝛼𝑝+2𝛽₂−𝐵₁−𝑏𝑝𝐵²₁(𝛼−1)𝛼𝑝+2𝛽2𝛼𝑝+𝛽²,𝜎₂=𝛼𝑝+𝛽²𝑏𝑝𝐵²₁×2𝐵𝛼𝑝+2𝛽₂+𝐵₁−𝑏𝑝𝐵²₁(𝛼−1)𝛼𝑝+2𝛽2𝛼𝑝+𝛽²,𝜎₃=𝛼𝑝+𝛽²𝑏𝑝𝐵²₁×𝛼𝑝+2𝛽2𝐵₂−𝑏𝑝𝐵²₁(𝛼−1)𝛼𝑝+2𝛽2𝛼𝑝+𝛽²,Φ𝑝,𝛼,𝛽,𝜇=𝛼𝑝+2𝛽2𝜇+𝛼−12𝛼𝑝+𝛽². (19) If 𝑓(𝑧) is of the form (1) and belongs to the class 𝑉_{𝑝,𝑏,𝛼,𝛽}(𝜙), then 𝑎_𝑝+2−𝜇𝑎²_𝑝+1≤𝑏𝑝2𝐵𝛼𝑝+2𝛽²−𝑏𝑝𝐵²₁2Φ𝑝,𝛼,𝛽,𝜇,𝜇<𝜎₁,𝑏𝑝𝐵₁2𝛼𝑝+2𝛽,𝜎₁≤𝜇≤𝜎₂,−𝑏𝑝2𝐵𝛼𝑝+2𝛽²−𝑏𝑝𝐵²₁2Φ𝑝,𝛼,𝛽,𝜇,𝜇>𝜎₂. (20) Furthermore, for 𝜎₁≤𝜇≤𝜎₃, 𝑎_𝑝+2−𝜇𝑎²_𝑝+1+1𝑏𝑝𝐵₁2𝐵1−²𝐵₁𝛼𝑝+𝛽²+𝛼𝑝+2𝛽𝑏𝑝𝐵₁22𝜇+𝛼−1𝑎_𝑝+1²≤𝑏𝑝𝐵₁2,𝛼𝑝+2𝛽 (21) and for 𝜎₃≤𝜇≤𝜎₂, 𝑎_𝑝+2−𝜇𝑎²_𝑝+1+1𝑏𝑝𝐵₁2𝐵1+²𝐵₁𝛼𝑝+𝛽²−𝛼𝑝+2𝛽𝑏𝑝𝐵₁22𝜇+𝛼−1𝑎_𝑝+1²≤𝑏𝑝𝐵₁2.𝛼𝑝+2𝛽 (22) For any complex number 𝜇, 𝑎_𝑝+2−𝜇𝑎²_𝑝+1≤𝑏𝑝𝐵₁2𝛼𝑝+2𝛽max1,𝑏𝑝𝐵₁2Φ−𝐵𝑝,𝛼,𝛽,𝜇²𝐵₁. (23) Also, 𝑎_𝑝+3≤𝑏𝑝𝐵₁2𝐻𝑞𝛼𝑝+3𝛽¹,𝑞₂, (24) where 𝐻(𝑞₁,𝑞₂) is defined in Lemma 3 and 𝑞₁=2𝐵₂𝐵₁+𝑏𝑝𝐵₁2(1−𝛼)𝛼𝑝+3𝛽,𝑞𝛼𝑝+𝛽𝛼𝑝+2𝛽₂=𝐵₃𝐵₁+𝑏𝑝𝐵₁2²(𝛼−1)(2𝛼−1)𝛼𝑝+3𝛽6𝛼𝑝+𝛽³+𝑏𝑝𝐵₂2(1−𝛼)𝛼𝑝+3𝛽.𝛼𝑝+𝛽𝛼𝑝+2𝛽 (25) These results are sharp.

Proof. Since 𝑓∈𝑉_{𝑝,𝑏,𝛼,𝛽}(𝜙), therefore we have for a Schwarz function 𝑤(𝑧)=𝑤₁𝑧+𝑤₂𝑧²+⋯,𝑧∈𝐸 (26) such that 21−𝑏+2𝑏1−𝛽𝑓(𝑧)𝑧^𝑝𝛼+𝛽𝑧𝑓^′(𝑧)𝑝𝑓(𝑧)𝑓(𝑧)𝑧^𝑝𝛼=𝜙(𝑤(𝑧)). (27) Now, 21−𝑏+2𝑏1−𝛽𝑓(𝑧)𝑧^𝑝𝛼+𝛽𝑧𝑓^′(𝑧)𝑝𝑓(𝑧)𝑓(𝑧)𝑧^𝑝𝛼2=1+𝑎𝑏𝑝𝛼𝑝+𝛽_𝑝+11𝑧+×𝑏𝑝𝛼𝑝+2𝛽2𝑎_𝑝+2+(𝛼−1)𝑎²_𝑝+1𝑧²+2𝛼𝑝+3𝛽×𝑎𝑏𝑝_𝑝+3+(𝛼−1)𝑎_𝑝+1𝑎_𝑝+2+(𝛼−1)(𝛼−2)6𝑎³_𝑝+1𝑧³+⋯. (28) Also, we have 𝜙(𝑤(𝑧))=1+𝐵₁𝑤₁𝑧+𝐵₁𝑤₂+𝐵₁𝑤²₁𝑧²+𝐵₁𝑤₃+2𝐵₂𝑤₁𝑤₂+𝐵₃𝑤³₁𝑧³+⋯. (29) Comparing the coefficients of 𝑧, 𝑧², 𝑧³ and after simple calculations, we obtain 𝑎_𝑝+1=𝑏𝑝𝐵₁𝑤₁2,𝑎𝛼𝑝+𝛽_𝑝+2=𝑏𝑝𝐵₁2×𝑤𝛼𝑝+2𝛽₂−𝑏𝑝𝐵₁2(𝛼−1)𝛼𝑝+2𝛽2𝛼𝑝+𝛽²−𝐵₂𝐵₁𝑤²₁,𝑎_𝑝+3=𝑏𝑝𝐵₁2𝑤𝛼𝑝+2𝛽³+𝑞₁𝑤₁𝑤₂+𝑞₂𝑤³₁, (30) where 𝑞₁ and 𝑞₂ are defined in (25). It can be easily followed from (30) that 𝑎_𝑝+2−𝜇𝑎²_𝑝+1=𝑏𝑝𝐵₁2𝑤𝛼𝑝+2𝛽²−𝜈𝑤²₁, (31) where 𝜈=𝑏𝑝𝐵₁2Φ−𝐵𝑝,𝛼,𝛽,𝜇²𝐵₁. (32) The results from (20) to (22)are obtained by using Lemma 3, (23)by using Lemma 4, and (24) by using Lemma 5. To show that these results are sharp, we define the functions 𝐾_𝜙𝑛(𝑧), 𝐹_𝜆(𝑧), and 𝐺_𝜆(𝑧) such that 𝐾_𝜙𝑛(0)=𝐾_𝜙𝑛^′(0)𝐹−1=0,_𝜆(0)=𝐹^′_𝜆(0)𝐺−1=0,_𝜆(0)=𝐺^′_𝜆(0)−1=0, (33) with 21−𝑏+2𝑏𝐾1−𝛽^𝜙𝑛(𝑧)𝑧^𝑝𝛼+𝛽𝑧𝐾^′_𝜙𝑛(𝑧)𝑝𝐾_𝜙𝑛(𝑧)𝐾_𝜙𝑛(𝑧)𝑧^𝑝𝛼≺𝜙𝑧^𝑛−1,21−𝑏+2𝑏𝐹1−𝛽^𝜆(𝑧)𝑧^𝑝𝛼+𝛽𝑧𝐹^′_𝜆(𝑧)𝑝𝐹_𝜆(𝑧)𝐹_𝜆(𝑧)𝑧^𝑝𝛼𝑧()≺𝜙𝑧+𝜆,21+𝜆𝑧1−𝑏+2𝑏𝐺1−𝛽^𝜆(𝑧)𝑧^𝑝𝛼+𝛽𝑧𝐺^′_𝜆(𝑧)𝑝𝐺_𝜆(𝑧)𝐺_𝜆(𝑧)𝑧^𝑝𝛼−𝑧()≺𝜙𝑧+𝜆.1+𝜆𝑧 (34) It is clear that the functions 𝐾_𝜙𝑛,𝐹_𝜆,𝐺_𝜆∈𝑉_{𝑝,𝑏,𝛼,𝛽}(𝜙). Let 𝐾_𝜙∶=𝐾_𝜙2. If 𝜇<𝜎₁ or 𝜇>𝜎₂, then the equality occurs for the function 𝐾_𝜙 or one of its rotations. For 𝜎₁<𝜇<𝜎₂, the equality is attained, if and only if 𝑓 is 𝐾_𝜙3 or one of its rotations. When 𝜇=𝜎₁, then the equality holds for the function 𝐹_𝜆 or one of its rotations. If 𝜇=𝜎₂, then the equality is obtained for the function 𝐺_𝜆 or one of its rotations.

Corollary 7. For 𝑏=2, the results from (20) to (24) coincide with the results proved by Ramachandran et al. [8].

Corollary 8. For 𝑏=2,  𝑝=1, and 𝛽=1,the results from (20) to (22) coincide with the results obtained by Ravichandran et al. [4] for the class 𝐵_𝛼(𝜙).

Corollary 9. For 𝑏=2,  𝛼=0, and 𝛽=1,the results from (20) to (24) coincide with the results obtained by Ali et al. [1] for the class 𝑆^∗_𝑝(𝜙).

Corollary 10. For 𝑏=2, 𝑝=1, 𝛼=0, and 𝛽=1,the results from (20) to (22) coincide with the results obtained by Ma and Minda [2] for the class 𝑆^∗(𝜙).

2.1. Application of Theorem 6 to the Function Defined by Convolutions

Theorem 11. Let 𝜙(𝑧)=1+𝐵₁𝑧+𝐵₂𝑧²+𝐵₃𝑧³+⋯, where 𝐵_𝑛, 𝑠 are real with 𝐵₁>0 and 𝐵₂≥0. Let 𝜎₁=𝑔²_𝑝+1𝑔_𝑝+2𝛼𝑝+𝛽²𝑏𝑝𝐵²₁×2𝐵𝛼𝑝+2𝛽₂−𝐵₁−𝑏𝑝𝐵²₁(𝛼−1)𝛼𝑝+2𝛽2𝛼𝑝+𝛽²,𝜎₂=𝑔²_𝑝+1𝑔_𝑝+2𝛼𝑝+𝛽²𝑏𝑝𝐵²₁×2𝐵𝛼𝑝+2𝛽₂+𝐵₁−𝑏𝑝𝐵²₁(𝛼−1)𝛼𝑝+2𝛽2𝛼𝑝+𝛽²,𝜎₃=𝑔²_𝑝+1𝑔_𝑝+2𝛼𝑝+𝛽²𝑏𝑝𝐵²₁×𝛼𝑝+2𝛽2𝐵₂−𝑏𝑝𝐵²₁(𝛼−1)𝛼𝑝+2𝛽2𝛼𝑝+𝛽²,Φ^∗=𝑔𝑝,𝛼,𝛽,𝜇𝛼𝑝+2𝛽2𝜇_𝑝+2/𝑔_𝑝+1+𝛼−12𝛼𝑝+𝛽². (35) If 𝑓(𝑧) is of the form (1) and belongs to the class 𝑉_{𝑝,𝑏,𝛼,𝛽,𝑔}(𝜙), then 𝑎_𝑝+2−𝜇𝑎²_𝑝+1≤𝑏𝑝2𝑔_𝑝+2𝐵𝛼𝑝+2𝛽²−𝑏𝑝𝐵²₁2Φ^∗𝑝,𝛼,𝛽,𝜇,𝜇<𝜎₁,𝑏𝑝𝐵₁2𝑔_𝑝+2𝛼𝑝+2𝛽,𝜎₁≤𝜇≤𝜎₂,𝑏𝑝2𝑔_𝑝+2𝛼𝑝+2𝛽𝑏𝑝𝐵²₁2Φ^∗𝑝,𝛼,𝛽,𝜇−𝐵₂,𝜇>𝜎₂. (36) Furthermore, for 𝜎₁≤𝜇≤𝜎₃, 𝑎_𝑝+2−𝜇𝑎²_𝑝+1+𝑔²_𝑝+1𝑏𝑝𝐵₁𝑔_𝑝+22𝐵1−²𝐵₁𝛼𝑝+𝛽²+𝛼𝑝+2𝛽𝑏𝑝𝐵₁2𝑔2𝜇^𝑝+2𝑔_𝑝+1+𝛼−1𝑎_𝑝+1²≤𝑏𝑝𝐵₁2𝑔_𝑝+2,𝛼𝑝+2𝛽 (37) and for 𝜎₃≤𝜇≤𝜎₂, 𝑎_𝑝+2−𝜇𝑎²_𝑝+1+𝑔²_𝑝+1𝑏𝑝𝐵₁𝑔_𝑝+22𝐵1+²𝐵₁𝛼𝑝+𝛽²−𝛼𝑝+2𝛽𝑏𝑝𝐵₁2𝑔2𝜇^𝑝+2𝑔_𝑝+1+𝛼−1𝑎_𝑝+1²≤𝑏𝑝𝐵₁2𝑔_𝑝+2.𝛼𝑝+2𝛽 (38) For any complex number 𝜇, 𝑎_𝑝+2−𝜇𝑎²_𝑝+1≤𝑏𝑝𝐵₁2𝑔_𝑝+2𝛼𝑝+2𝛽max1,𝑏𝑝𝐵₁2Φ^∗−𝐵𝑝,𝛼,𝛽,𝜇²𝐵₁. (39) Also, 𝑎_𝑝+3≤𝑏𝑝𝐵₁2𝛼𝑝+3𝛽𝑔_𝑝+3𝐻𝑞₁,𝑞₂, (40) where 𝐻(𝑞₁,𝑞₂)is defined in Lemma 5 and 𝑞₁=2𝐵₂𝐵₁+𝑏𝑝𝐵₁2(1−𝛼)𝛼𝑝+3𝛽,𝑞𝛼𝑝+𝛽𝛼𝑝+2𝛽₂=𝐵₃𝐵₁+𝑏𝑝𝐵₁2²(𝛼−1)(2𝛼−1)𝛼𝑝+3𝛽6𝛼𝑝+𝛽³+𝑏𝑝𝐵₂2(1−𝛼)𝛼𝑝+3𝛽.𝛼𝑝+𝛽𝛼𝑝+2𝛽 (41) These results are sharp.

Proof. Since 𝑓∈𝑉_{𝑝,𝑏,𝛼,𝛽,𝑔}(𝜙), therefore we have for a Schwarz function 𝑤,such that 21−𝑏+2𝑏1−𝛽𝑓∗𝑔(𝑧)𝑧^𝑝𝛼𝑧+𝛽𝑓∗𝑔^′(𝑧)𝑝𝑓∗𝑔(𝑧)𝑓∗𝑔(𝑧)𝑧^𝑝𝛼=𝜙(𝑤(𝑧)). (42) Now, 21−𝑏+2𝑏1−𝛽𝑓∗𝑔(𝑧)𝑧^𝑝𝛼𝑧+𝛽𝑓∗𝑔^′(𝑧)𝑝𝑓∗𝑔(𝑧)𝑓∗𝑔(𝑧)𝑧^𝑝𝛼2=1+𝑎𝑏𝑝𝛼𝑝+𝛽_𝑝+1𝑔_𝑝+11𝑧+×𝑏𝑝𝛼𝑝+2𝛽2𝑎_𝑝+2𝑔_𝑝+2+(𝛼−1)𝑎²_𝑝+1𝑔²_𝑝+1𝑧²+2×𝑎𝑏𝑝𝛼𝑝+3𝛽_𝑝+3𝑔_𝑝+3+(𝛼−1)𝑎_𝑝+1𝑎_𝑝+2𝑔_𝑝+1𝑔_𝑝+2+(𝛼−1)(𝛼−2)6𝑎³_𝑝+1𝑔³_𝑝+1𝑧³+⋯. (43) Also, we obtain 𝜙(𝑤(𝑧))=1+𝐵₁𝑤₁𝑧+𝐵₁𝑤₂+𝐵₁𝑤²₁𝑧²+𝐵₁𝑤₃+2𝐵₂𝑤₁𝑤₂+𝐵₃𝑤³₁𝑧³+⋯. (44) Comparing the coefficients of 𝑧, 𝑧², 𝑧³ and after simple calculations, we obtain 𝑎_𝑝+1=𝑏𝑝𝐵₁𝑤₁2𝑔_𝑝+1,𝑎𝛼𝑝+𝛽_𝑝+2=𝑏𝑝𝐵₁2𝑔_𝑝+2×𝑤𝛼𝑝+2𝛽₂−𝑏𝑝𝐵₁2(𝛼−1)𝛼𝑝+2𝛽2𝛼𝑝+𝛽²−𝐵₂𝐵₁𝑤²₁,𝑎_𝑝+3=𝑏𝑝𝐵₁2𝑔_𝑝+3𝑤𝛼𝑝+2𝛽³+𝑞₁𝑤₁𝑤₂+𝑞₂𝑤³₁. (45) The remaining proof of the theorem is similar to the proof of Theorem 6.

Corollary 12. For 𝑏=2, the results from (36) to (40) coincide with the results proved by Ramachandran et al. [8] for the class 𝑅_{𝑝,1,𝛼,𝛽,𝑔}(𝜙).

Corollary 13. For 𝑏=2, 𝛼=0, and 𝛽=1,the results from (36) to (40) coincide with the results obtained by Ali et al. [1] for the class 𝑆^∗_𝑝,𝑔(𝜙).

Corollary 14. For 𝑏=2, 𝑝=1, 𝛼=0, and 𝛽=1, 𝑔₂=Γ(3)Γ(2−𝜆)Γ()=23−𝜆,𝑔2−𝜆₃=Γ(4)Γ(2−𝜆)Γ()=64−𝜆(),𝐵2−𝜆)(3−𝜆₁=8𝜋²,𝐵₂=163𝜋², (46) the results from (36) to (38) coincide with the results obtained by Srivastava and Mishra [9].

Acknowledgment

The work here is fully supported by LRGS/TD/2011/UKM/ICT/03/02.