Coefficient Estimates for New Subclasses of Analytic and Bi-Univalent Functions Defined by Al-Oboudi Differential Operator

Abstract

We introduce and investigate two new subclasses 𝒩_Σ^𝛿,𝜇(𝑛,𝛼,𝜆) and 𝒩_Σ^𝛿,𝜇(𝑛,𝛽,𝜆) of analytic and bi-univalent functions in the open unit disk 𝕌. For functions belonging to these classes, we obtain estimates on the first two Taylor-Maclaurin coefficients |𝑎₂| and |𝑎₃|.

1. Introduction

Let ℝ=(−∞,∞) be the set of real numbers, ℂ the set of complex numbers, and ℕ∶={1,2,3,…}=ℕ₀⧵{0} (1) the set of positive integers.

Let 𝒜 denote the class of all functions of the form 𝑓(𝑧)=𝑧+^∞_𝑘=2𝑎_𝑘𝑧^𝑘 (2) which are analytic in the open unit disk 𝕌={𝑧∶𝑧∈ℂ,|𝑧|<1}. (3) We also denote by 𝒮 the class of all functions in the normalized analytic function class 𝒜 which are univalent in 𝕌.

For 𝑓∈𝒜, Al-Oboudi [1] introduced the following operator: 𝐷⁰_𝛿𝑓(𝑧)(𝑧),𝐷=𝑓_1𝛿𝑓(𝑧)=(1−𝛿)𝑓(𝑧)+𝛿𝑧𝑓^′(𝑧)=∶𝐷_𝛿𝑓(𝑧)𝐷)(𝛿≥0_𝑛𝛿𝑓(𝑧)=𝐷_𝛿𝐷_𝛿^𝑛−1𝑓(𝑧)(𝑛∈ℕ). (4)(5)(6) If 𝑓 is given by (2), then from (5) and (6) we see that 𝐷^𝑛_𝛿𝑓(𝑧)=𝑧+^∞_𝑘=2[1+(𝑘−1)𝛿]^𝑛𝑎_𝑘𝑧^𝑘𝑛∈ℕ₀, (7) with 𝐷^𝑛_𝛿𝑓(0)=0. When 𝛿=1, we get Sălăgean’s differential operator [2].

Since univalent functions are one-to-one, they are invertible and the inverse functions need not be defined on the entire unit disk 𝕌. In fact, the Koebe one-quarter theorem [3] ensures that the image of 𝕌 under every univalent function 𝑓∈𝒮 contains a disk of radius 1/4. Thus, every function 𝑓∈𝒜 has an inverse 𝑓⁻¹, which is defined by 𝑓⁻¹𝑓(𝑧(),𝑓𝑓)=𝑧𝑧∈𝕌₋₁(𝑤)=𝑤|𝑤|<𝑟₀𝑓;𝑟₀𝑓≥14. (8) In fact, the inverse function 𝑓⁻¹ is given by 𝑓⁻¹(𝑤)=𝑤−𝑎₂𝑤²+2𝑎²₂−𝑎₃𝑤³−5𝑎³₂−5𝑎₂𝑎₃+𝑎₄𝑤⁴+⋯. (9)

A function 𝑓∈𝒜 is said to be bi-univalent in 𝕌 if both 𝑓 and 𝑓⁻¹ are univalent in 𝕌. Let Σ denote the class of bi-univalent functions in 𝕌 given by (2). For a brief history and interesting examples of functions in the class Σ, see [4] (see also [5, 6]). In fact, the aforecited work of Srivastava et al. [4] essentially revived the investigation of various subclasses of the bi-univalent function class Σ in recent years; it was followed by such works as those by Frasin and Aouf [7], Çağlar et al. [8], Porwal and Darus [9], and others (see, for example, [10–13]).

The object of the present paper is to introduce two new subclasses of the function class Σ and find estimates on the coefficients |𝑎₂| and |𝑎₃| for functions in these new subclasses of the function class Σ.

Firstly, in order to derive our main results, we need the following lemma.

Lemma 1 (see [14]). If 𝑝∈𝒫, then |𝑐_𝑘|≤2 for each 𝑘, where 𝒫 is the family of all functions 𝑝 analytic in 𝕌 for which ℜ𝑝(𝑧)>0,𝑝(𝑧)=1+𝑐₁𝑧+𝑐₂𝑧²+⋯ (10) for 𝑧∈𝕌.

2. Coefficient Bounds for the Function Class 𝒩_Σ^𝛿,𝜇(𝑛,𝛼,𝜆)

Definition 2. A function 𝑓(𝑧) given by (2) is said to be in the class 𝒩_Σ^𝛿,𝜇(𝑛,𝛼,𝜆) if the following conditions are satisfied: ()𝐷𝑓∈Σ,arg1−𝜆^𝑛𝛿𝑓(𝑧)𝑧^𝜇+𝜆𝐷^𝑛_𝛿𝑓(𝑧)^′𝐷^𝑛_𝛿𝑓(𝑧)𝑧^𝜇−1<𝛼𝜋20<𝛼≤1,𝜆≥1,𝜇≥0,𝛿≥0,𝑛∈ℕ₀,()𝐷,𝑧∈𝕌arg1−𝜆^𝑛𝛿𝑔(𝑤)𝑤^𝜇+𝜆𝐷^𝑛_𝛿𝑔(𝑤)^′𝐷^𝑛_𝛿𝑔(𝑤)𝑤^𝜇−1<𝛼𝜋20<𝛼≤1,𝜆≥1,𝜇≥0,𝛿≥0,𝑛∈ℕ₀,𝑤∈𝕌, (11) where the function 𝑔 is given by 𝑔(𝑤)=𝑤−𝑎₂𝑤²+2𝑎²₂−𝑎₃𝑤³−5𝑎³₂−5𝑎₂𝑎₃+𝑎₄𝑤⁴+⋯ (12) and 𝐷^𝑛_𝛿 is the Al-Oboudi differential operator.

Remark 3. In Definition 2, if we choose(i)𝑛=0, then we have the class 𝒩_Σ^𝛿,𝜇(0,𝛼,𝜆)=𝒩^𝜇_Σ(𝛼,𝜆) (13) introduced by Çağlar et al. [8];(ii)𝑛=0 and 𝜇=1, then we have the class 𝒩_Σ^𝛿,1(0,𝛼,𝜆)=ℬ_Σ(𝛼,𝜆) (14) introduced by Frasin and Aouf [7];(iii)𝑛=0,𝜇=1, and 𝜆=1, then we have the class 𝒩_Σ^𝛿,1(0,𝛼,1)=ℋ^𝛼_Σ (15) introduced by Srivastava et al. [4];(iv)𝑛=0,𝜇=0, and 𝜆=1, then we have the class 𝒩_Σ^𝛿,0(0,𝛼,1)=𝒮^∗_Σ[𝛼] (16) of strongly bi-starlike functions of order 𝛼, introduced by Brannan and Taha [5, 6];(v)𝛿=1 and 𝜇=1, then we have the class 𝒩_Σ^1,1(𝑛,𝛼,𝜆)=ℬ_Σ(𝑛,𝛼,𝜆) (17) introduced by Porwal and Darus [9].

Theorem 4. Let the function 𝑓(𝑧) given by the Taylor-Maclaurin series expansion (2) be in the function class 𝒩_Σ^𝛿,𝜇(𝑛,𝛼,𝜆)0<𝛼≤1,𝜆≥1,𝜇≥0,𝛿≥0,𝑛∈ℕ₀. (18) Then, 𝑎₂≤()×()2𝛼1+𝛿^2𝑛𝜆+𝜇²+𝛼2(1+2𝛿)^𝑛−()2𝜆+𝜇1+𝛿^2𝑛𝜆²+2𝜆+𝜇^1/2−1𝑎₃≤4𝛼²(1+𝛿)^2𝑛𝜆+𝜇²+2𝛼(1+2𝛿)^𝑛.2𝜆+𝜇 (19)(20)

Proof. First of all, it follows from conditions (11) that ()𝐷1−𝜆^𝑛𝛿𝑓(𝑧)𝑧^𝜇+𝜆𝐷^𝑛_𝛿𝑓(𝑧)^′𝐷^𝑛_𝛿𝑓(𝑧)𝑧^𝜇−1=𝑝(𝑧)^𝛼(),()𝐷𝑧∈𝕌1−𝜆^𝑛𝛿𝑔(𝑤)𝑤^𝜇+𝜆𝐷^𝑛_𝛿𝑔(𝑤)^′𝐷^𝑛_𝛿𝑔(𝑤)𝑤^𝜇−1=𝑞(𝑤)^𝛼(𝑤∈𝕌), (21) respectively, where 𝑝(𝑧)=1+𝑝₁𝑧+𝑝₂𝑧²𝑞(𝑤)+⋯,=1+𝑞₁𝑤+𝑞₂𝑤²+⋯ (22) in 𝒫. Now, upon equating the coefficients in (21), we get (1+𝛿)^𝑛𝜆+𝜇𝑎₂=𝛼𝑝₁,(1+2𝛿)^𝑛2𝜆+𝜇𝑎₃+(1+𝛿)^2𝑛𝜇𝜇−1𝜆+2𝑎²₂=𝛼𝑝₂+𝛼(𝛼−1)2𝑝²₁,−(1+𝛿)^𝑛𝜆+𝜇𝑎₂=𝛼𝑞₁,−(1+2𝛿)^𝑛2𝜆+𝜇𝑎₃+4(1+2𝛿)^𝑛+(1+𝛿)^2𝑛×𝜇𝜇−1𝜆+2𝑎²₂=𝛼𝑞₂+𝛼(𝛼−1)2𝑞²₁. (23)(24)(25)(26) From (23) and (25), we obtain 𝑝₁=−𝑞₁,2(1+𝛿)^2𝑛𝜆+𝜇²𝑎²₂=𝛼²𝑝²₁+𝑞²₁. (27)(28) Also, from (24), (26), and (28), we find that 2(1+2𝛿)^𝑛+(1+𝛿)^2𝑛𝜇−12𝜆+𝜇𝑎²₂=𝛼𝑝₂+𝑞₂+𝛼(𝛼−1)2𝑝²₁+𝑞²₁=𝛼𝑝₂+𝑞₂+𝛼−1𝛼(1+𝛿)^2𝑛𝜆+𝜇²𝑎²₂. (29) Therefore, we obtain 𝑎²₂=𝛼²𝑝₂+𝑞₂×2𝛼(1+2𝛿)^𝑛+()2𝜆+𝜇1+𝛿^2𝑛𝜆+𝜇²−𝛼𝜆²+2𝜆+𝜇⁻¹. (30) Applying Lemma 1 for the aforementioned equality, we get the desired estimate on the coefficient |𝑎₂| as asserted in (19).
Next, in order to find the bound on the coefficient |𝑎₃|, we subtract (26) from (24). We thus get 2(1+2𝛿)^𝑛2𝜆+𝜇𝑎₃−2(1+2𝛿)^𝑛2𝜆+𝜇𝑎²₂=𝛼𝑝₂−𝑞₂+𝛼(𝛼−1)2𝑝²₁−𝑞²₁. (31) It follows from (27), (28), and (31) that 𝑎₃=𝛼²𝑝²₁+𝑞²₁2(1+𝛿)^2𝑛𝜆+𝜇²+𝛼𝑝₂−𝑞₂2(1+2𝛿)^𝑛.2𝜆+𝜇 (32) Applying Lemma 1 for the previous equality, we get the desired estimate on the coefficient |𝑎₃| as asserted in (20).

If we take 𝑛=0 in Theorem 4, then we have the following corollary.

Corollary 5 (see [8]). Let the function 𝑓(𝑧) given by the Taylor-Maclaurin series expansion (2) be in the class 𝒩^𝜇_Σ(𝛼,𝜆)(0<𝛼≤1,𝜆≥1,𝜇≥0). Then, 𝑎₂≤2𝛼𝜆+𝜇²+𝛼𝜇+2𝜆−𝜆²,𝑎₃≤4𝛼²𝜆+𝜇²+2𝛼.2𝜆+𝜇 (33)

If we take 𝑛=0 and 𝜇=1 in Theorem 4, then we have the following corollary.

Corollary 6 (see [7]). Let the function 𝑓(𝑧) given by the Taylor-Maclaurin series expansion (2) be in the class ℬ_Σ(𝛼,𝜆)(0<𝛼≤1,𝜆≥1). Then 𝑎₂≤2𝛼(𝜆+1)²+𝛼1+2𝜆−𝜆²,𝑎₃≤4𝛼²(𝜆+1)²+2𝛼.2𝜆+1 (34)

If we take 𝑛=0,𝜇=1 and 𝜆=1 in Theorem 4, then we have the following corollary.

Corollary 7 (see [4]). Let the function 𝑓(𝑧) given by the Taylor-Maclaurin series expansion (2) be in the class ℋ^𝛼_Σ(0<𝛼≤1). Then, 𝑎₂≤𝛼2,𝑎𝛼+2₃≤𝛼(3𝛼+2)3. (35)

If we take 𝑛=0,𝜇=0, and 𝜆=1 in Theorem 4, then we have the following corollary.

Corollary 8. Let the function 𝑓(𝑧) given by the Taylor-Maclaurin series expansion (2) be in the class 𝒮^∗_Σ[𝛼](0<𝛼≤1). Then, 𝑎₂≤2𝛼√,𝑎1+𝛼₃≤4𝛼²+𝛼. (36)

If we take 𝛿=1 and 𝜇=1 in Theorem 4, then we have the following corollary.

Corollary 9 (see [9]). Let the function 𝑓(𝑧) given by the Taylor-Maclaurin series expansion (2) be in the function class ℬ_Σ(𝑛,𝛼,𝜆)(0<𝛼≤1,𝜆≥1,𝑛∈ℕ₀). Then, 𝑎₂≤2𝛼4^𝑛(𝜆+1)²+𝛼2.3^𝑛(2𝜆+1)−4^𝑛(𝜆+1)²,𝑎₃≤4𝛼²4^𝑛(𝜆+1)²+2𝛼3^𝑛().2𝜆+1 (37)

3. Coefficient Bounds for the Function Class 𝒩_Σ^𝛿,𝜇(𝑛,𝛽,𝜆)

Definition 10. A function 𝑓(𝑧) given by (2) is said to be in the class 𝒩_Σ^𝛿,𝜇(𝑛,𝛽,𝜆) if the following conditions are satisfied: ℜ)𝐷𝑓∈Σ,(1−𝜆^𝑛𝛿𝑓(𝑧)𝑧^𝜇+𝜆𝐷^𝑛_𝛿𝑓(𝑧)^′𝐷^𝑛_𝛿𝑓(𝑧)𝑧^𝜇−1>𝛽0≤𝛽<1,𝜆≥1,𝜇≥0,𝛿≥0,𝑛∈ℕ₀,ℜ)𝐷,𝑧∈𝕌(1−𝜆^𝑛𝛿𝑔(𝑤)𝑤^𝜇+𝜆𝐷^𝑛_𝛿𝑔(𝑤)^′𝐷^𝑛_𝛿𝑔(𝑤)𝑤^𝜇−1>𝛽0≤𝛽<1,𝜆≥1,𝜇≥0,𝛿≥0,𝑛∈ℕ₀,𝑤∈𝕌, (38) where the function 𝑔 is defined by (12) and 𝐷^𝑛_𝛿 is the Al-Oboudi differential operator.

Remark 11. In Definition 10, if we choose(i)𝑛=0, then we have the class 𝒩_Σ^𝛿,𝜇0,𝛽,𝜆=𝒩^𝜇_Σ𝛽,𝜆 (39) introduced by Çağlar et al. [8];(ii)𝑛=0 and 𝜇=1, then we have the class 𝒩_Σ^𝛿,10,𝛽,𝜆=ℬ_Σ𝛽,𝜆 (40) introduced by Frasin and Aouf [7];(iii)𝑛=0,𝜇=1, and 𝜆=1, then we have the class 𝒩_Σ^𝛿,10,𝛽,1=ℋ_Σ𝛽 (41) introduced by Srivastava et al. [4];(iv)𝑛=0,𝜇=0, and 𝜆=1, then we have the class 𝒩_Σ^𝛿,00,𝛽,1=𝒮^∗_Σ𝛽 (42) of bi-starlike functions of order 𝛽, introduced by Brannan and Taha [5, 6];(v)𝛿=1 and 𝜇=1, then we have the class 𝒩_Σ^1,1𝑛,𝛽,𝜆=ℋ_Σ𝑛,𝛽,𝜆 (43) introduced by Porwal and Darus [9].

Theorem 12. Let the function 𝑓(𝑧) given by the Taylor-Maclaurin series expansion (2) be in the function class 𝒩_Σ^𝛿,𝜇𝑛,𝛽,𝜆0<𝛼≤1,𝜆≥1,𝜇≥0,𝛿≥0,𝑛∈ℕ₀. (44) Then, 𝑎₂2≤min1−𝛽(1+𝛿)^𝑛,4×2()𝜆+𝜇1−𝛽1+2𝛿^𝑛+(1+𝛿)^2𝑛×𝜇−12𝜆+𝜇^−11/2𝑎₃4≤min1−𝛽²(1+𝛿)^2𝑛𝜆+𝜇²+21−𝛽(1+2𝛿)^𝑛,4()2𝜆+𝜇1−𝛽1+2𝛿^𝑛+(1+𝛿)^2𝑛+()𝜇−11+𝛿^2𝑛×()𝜇−11+2𝛿^𝑛2(1+2𝛿)^𝑛+(1+𝛿)^2𝑛×𝜇−12𝜆+𝜇⁻¹. (45)(46)

Proof. First of all, it follows from conditions (38) that ()𝐷1−𝜆^𝑛𝛿𝑓(𝑧)𝑧^𝜇+𝜆𝐷^𝑛_𝛿𝑓(𝑧)^′𝐷^𝑛_𝛿𝑓(𝑧)𝑧^𝜇−1𝑝(𝑧),()𝐷=𝛽+1−𝛽)(𝑧∈𝕌1−𝜆^𝑛𝛿𝑔(𝑤)𝑤^𝜇+𝜆𝐷^𝑛_𝛿𝑔(𝑤)^′𝐷^𝑛_𝛿𝑔(𝑤)𝑤^𝜇−1=𝛽+1−𝛽𝑞(𝑤)(𝑤∈𝕌), (47) respectively, where 𝑝(𝑧)=1+𝑝₁𝑧+𝑝₂𝑧²𝑞(𝑤)+⋯,=1+𝑞₁𝑤+𝑞₂𝑤²+⋯ (48) in 𝒫. Now, upon equating the coefficients in (47), we get (1+𝛿)^𝑛𝜆+𝜇𝑎₂=1−𝛽𝑝₁,(1+2𝛿)^𝑛2𝜆+𝜇𝑎₃+(1+𝛿)^2𝑛𝜇𝜇−1𝜆+2𝑎²₂=1−𝛽𝑝₂,−(1+𝛿)^𝑛𝜆+𝜇𝑎₂=1−𝛽𝑞₁,−(1+2𝛿)^𝑛2𝜆+𝜇𝑎₃+4(1+2𝛿)^𝑛+(1+𝛿)^2𝑛×𝜇𝜇−1𝜆+2𝑎²₂=1−𝛽𝑞₂. (49)(50)(51)(52) From (49) and (51), we obtain 𝑝₁=−𝑞₁,2(1+𝛿)^2𝑛𝜆+𝜇²𝑎²₂=1−𝛽²𝑝²₁+𝑞²₁. (53)(54) Also, from (50) and (52), we have 2(1+2𝛿)^𝑛+(1+𝛿)^2𝑛×𝑎𝜇−12𝜆+𝜇²₂=1−𝛽𝑝₂+𝑞₂. (55) Therefore, from equalities (54) and (55) we find that 𝑎₂²≤1−𝛽²𝑝₁²+𝑞₁²2(1+𝛿)^2𝑛𝜆+𝜇²,𝑎₂²≤1−𝛽𝑝₂+𝑞₂2(1+2𝛿)^𝑛+(1+𝛿)^2𝑛,𝜇−12𝜆+𝜇 (56) respectively, and applying Lemma 1, we get the desired estimate on the coefficient |𝑎₂| as asserted in (45).
Next, in order to find the bound on the coefficient |𝑎₃|, we subtract (52) from (50). We thus get 2(1+2𝛿)^𝑛2𝜆+𝜇𝑎₃−2(1+2𝛿)^𝑛2𝜆+𝜇𝑎²₂=1−𝛽𝑝₂−𝑞₂, (57) which, upon substitution of the value of 𝑎²₂ from (54), yields 𝑎₃=1−𝛽²𝑝²₁+𝑞²₁2(1+𝛿)^2𝑛𝜆+𝜇²+1−𝛽𝑝₂−𝑞₂2(1+2𝛿)^𝑛.2𝜆+𝜇 (58) On the other hand, by using the (55) into (57), it follows that 𝑎₃=×2()1−𝛽1+2𝛿^𝑛2(1+2𝛿)^𝑛+(1+𝛿)^2𝑛×𝜇−12𝜆+𝜇⁻¹×4(1+2𝛿)^𝑛+(1+𝛿)^2𝑛𝜇−1𝑝₂−(1+𝛿)^2𝑛𝜇−1𝑞₂. (59) Applying Lemma 1 for (58) and (59), we get the desired estimate on the coefficient |𝑎₃| as asserted in (46).

If we take 𝑛=0 in Theorem 12, then we have the following corollary.

Corollary 13 (see [8]). Let the function 𝑓(𝑧) given by the Taylor-Maclaurin series expansion (2) be in the class 𝒩^𝜇_Σ(𝛽,𝜆)(0≤𝛽<1,𝜆≥1,𝜇≥0). Then, 𝑎₂≤min41−𝛽,2𝜇+12𝜆+𝜇1−𝛽,𝑎𝜆+𝜇₃≤4min1−𝛽,4𝜇+12𝜆+𝜇1−𝛽²𝜆+𝜇²+21−𝛽,22𝜆+𝜇0≤𝜇<11−𝛽,2𝜆+𝜇𝜇≥1. (60)

If we take 𝑛=0 and 𝜇=1 in Theorem 12, then we have the following corollary.

Corollary 14. Let the function 𝑓(𝑧) given by the Taylor-Maclaurin series expansion (2) be in the class ℬ_Σ(𝛽,𝜆)(0≤𝛽<1,𝜆≥1). Then, 𝑎₂≤min21−𝛽,22𝜆+11−𝛽,𝑎𝜆+1₃≤21−𝛽.2𝜆+1 (61)

Remark 15. Corollary 14 provides an improvement of the following estimates obtained by Frasin and Aouf [7].

Corollary 16 (see [7]). Let the function 𝑓(𝑧) given by the Taylor-Maclaurin series expansion (2) be in the class ℬ_Σ(𝛽,𝜆)(0≤𝛽<1,𝜆≥1). Then, 𝑎₂≤21−𝛽,𝑎2𝜆+1₃≤41−𝛽²(𝜆+1)²+21−𝛽.2𝜆+1 (62)

If we take 𝑛=0,𝜇=1, and 𝜆=1 in Theorem 12, then we have the following corollary.

Corollary 17. Let the function 𝑓(𝑧) given by the Taylor-Maclaurin series expansion (2) be in the class ℋ_Σ(𝛽)(0≤𝛽<1). Then, 𝑎₂≤21−𝛽31,0≤𝛽≤311−𝛽,3𝑎≤𝛽<1,₃≤21−𝛽3. (63)

Remark 18. Corollary 17 provides an improvement of the following estimates obtained by Srivastava et al. [4].

Corollary 19 (see [4]). Let the function 𝑓(𝑧) given by the Taylor-Maclaurin series expansion (2) be in the class ℋ_Σ(𝛽)(0≤𝛽<1). Then, 𝑎₂≤21−𝛽3,𝑎₃≤1−𝛽5−3𝛽3. (64)(65)

If we take 𝑛=0,𝜇=0, and 𝜆=1 in Theorem 12, then we have the following corollary.

Corollary 20. Let the function 𝑓(𝑧) given by the Taylor-Maclaurin series expansion (2) be in the class 𝒮^∗_Σ(𝛽)(0≤𝛽<1). Then, 𝑎₂≤2,𝑎1−𝛽₃≤231−𝛽,0≤𝛽≤4,31−𝛽5−4𝛽4≤𝛽<1. (66)

Remark 21. Corollary 20 provides an improvement of the following estimates obtained by Brannan and Taha [5, 6] (see also [10, Corollary 3.2]).

Corollary 22 (see [5, 6]). Let the function 𝑓(𝑧) given by the Taylor-Maclaurin series expansion (2) be in the class 𝒮^∗_Σ(𝛽)(0≤𝛽<1). Then, 𝑎₂≤2,𝑎1−𝛽₃≤21−𝛽. (67)

If we take 𝛿=1 and 𝜇=1 in Theorem 12, then we have the following corollary.

Corollary 23. Let the function 𝑓(𝑧) given by the Taylor-Maclaurin series expansion (2) be in the function class ℋ_Σ(𝑛,𝛽,𝜆)(0<𝛼≤1,𝜆≥1,𝑛∈ℕ₀). Then, 𝑎₂2≤min1−𝛽2^𝑛(),𝜆+121−𝛽3^𝑛(),𝑎2𝜆+1₃≤21−𝛽3^𝑛().2𝜆+1 (68)

Remark 24. Corollary 23 provides an improvement of the following estimates obtained by Porwal and Darus [9].

Corollary 25 (see [9]). Let the function 𝑓(𝑧) given by the Taylor-Maclaurin series expansion (2) be in the function class ℋ_Σ(𝑛,𝛽,𝜆)(0<𝛼≤1,𝜆≥1,𝑛∈ℕ₀). Then, 𝑎₂≤21−𝛽3^𝑛(),𝑎2𝜆+1₃≤41−𝛽²4^𝑛(𝜆+1)²+21−𝛽3^𝑛().2𝜆+1 (69)