Coefficient Estimate of Bi-univalent Functions ... - Semantic Scholar

Southeast Asian Bulletin of Mathematics (2014) 38: 433–444

Southeast Asian Bulletin of Mathematics c SEAMS. 2014 ⃝

Coefficient Estimate of Bi-univalent Functions Involving Caputo Fractional Calculus Operator G. Murugusundaramoorthy and K. Thilagavathi School of Advanced Sciences, VIT University, Vellore - 632014, India Email: [email protected]; [email protected]

Received 1 January 2013 Accepted 13 June 2013 Communicated by H.M. Srivastava AMS Mathematics Subject Classification(2000): 30C45 Abstract. In this paper, we estimate the second and third Maclaurin’s coefficients of certain subclass of bi-univalent functions in open unit disk defined by Caputo Fractional calculus operator and also indicate certain special cases. Keywords: Univalent functions; Bi-univalent functions; Coefficient estimates; Fractional calculus operator; Subordination.

1. Introduction and Definitions Let 𝒜 denote the class of functions of the form 𝑓 (𝑧) = 𝑧 +

∞ ∑

𝑎𝑛 𝑧 𝑛

(1)

𝑛=2

which are analytic and univalent in the open disc 𝕌 = {𝑧 : 𝑧 ∈ ℂ and ∣𝑧∣ < 1}. Further, by 𝒮 we shall denote the class of all functions 𝑓 (𝑧) in 𝒜 which are univalent in 𝕌 and indeed normalized by 𝑓 (0) = 𝑓 ′ (0) − 1 = 0. Some of the important and well-investigated subclasses of the univalent function class 𝒮 include (for example) the class 𝒮 ∗ (𝛼)(0 ≤ 𝛼 < 1) of starlike functions of order 𝛼 in 𝕌 and the class 𝒦(𝛼)(0 ≤ 𝛼 < 1) of convex functions of order 𝛼 in 𝕌.

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It is well known that every function 𝑓 ∈ 𝒮 has an inverse 𝑓 −1 defined by 𝑓 −1 (𝑓 (𝑧)) = 𝑧, 𝑓 (𝑓

−1

(𝑤)) = 𝑤

(𝑧 ∈ 𝕌), (∣𝑤∣ < 𝑟0 (𝑓 ), 𝑟0 (𝑓 ) ≥

1 ). 4

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 defined in the unit disk 𝕌. Since 𝑓 ∈ Σ has the Maclaurian series given by (1), a computation shows that its inverse 𝑔 = 𝑓 −1 has the expansion 𝑔(𝑤) = 𝑓 −1 (𝑤) = 𝑤 − 𝑎2 𝑤2 + (2𝑎22 − 𝑎3 )𝑤3 + ⋅ ⋅ ⋅ .

(2)

An analytic function 𝜙 is subordinate to another analytic function 𝜓, written as follows: 𝜙(𝑧) ≺ 𝜓(𝑧), (𝑧 ∈ 𝕌) provided that there exists an analytic function (that is, Schwarz function) 𝜔(𝑧), analytic in 𝕌, with 𝜔(0) = 0 and ∣𝜔(𝑧)∣ < 1, (𝑧 ∈ 𝕌) such that 𝜙(𝑧) = 𝜓(𝜔(𝑧)), (𝑧 ∈ 𝕌). In particular, if the function 𝜓 is univalent in 𝕌, the above subordination is equivalent to 𝜙(0) = 𝜓(0) and 𝜙(𝕌) ≺ 𝜓(𝕌). Ma and Minda [9] gave a unified presentation of various subclasses of starlike and convex functions for which either of the quantity 𝑧𝑓 ′ (𝑧) 𝑧𝑓 ′′ (𝑧) or 1 + ′ 𝑓 (𝑧) 𝑓 (𝑧) is subordinate to a more general superordinate function. For this purpose, they considered an analytic function 𝜙 with positive real part in the unit disk 𝕌, with 𝜙(0) = 1 and 𝜙′ (0) > 0 and 𝜙 maps 𝕌 onto a region starlike with respect to 1 and symmetric with respect to the real axis. The class of Ma-Minda starlike functions consists of functions 𝑓 ∈ 𝒜 satisfying the following subordination condition: 𝑧 𝑓 ′ (𝑧) ≺ 𝜙(𝑧), (𝑧 ∈ 𝕌). 𝑓 (𝑧) Similarly, the class of Ma-Minda convex functions in 𝕌 consists of functions 𝑓 ∈ 𝒜 satisfying the following subordination condition: 1+

𝑧 𝑓 ′′ (𝑧) ≺ 𝜙(𝑧), (𝑧 ∈ 𝕌). 𝑓 ′ (𝑧)

Coefficient Estimate of Bi-univalent Functions

435

A function 𝑓 is said to be bi-starlike of Ma-Minda type in 𝕌 or bi-convex of Ma-Minda type in 𝕌 if both 𝑓 and 𝑓 −1 are, respectively, Ma-Minda starlike in 𝕌 or Ma-Minda convex in 𝕌. These function classes are denoted by 𝒮Σ∗ (𝛼) and 𝒦Σ (𝛼) respectively. In the sequel, it is assumed that 𝜙 is an analytic function with positive real part in 𝕌 such that 𝜙(0) = 1, 𝜙′ (0) > 0 and 𝜙(𝕌) is symmetric with respect to the real axis. Such a function has a series expansion of the following form: 𝜙(𝑧) = 1 + 𝑐1 𝑧 + 𝑐2 𝑧 2 + 𝑐3 𝑧 3 + ..., (𝑐1 > 0, 𝑧 ∈ 𝕌). Recently, fractional calculus operators have found interesting application in the theory of analytic functions. The classical definition of fractional calculus and their other generalizations have fruitfully been applied in obtaining the characterization properties, coefficient estimates and distortion inequalities for various subclasses of analytic functions (see [12, 13, 19]). There are many definitions of fractional integration and differentiation (see [11, 17, 23]) can be found in various books (see [7, 10, 20, 26]). We recall the following definitions due to Owa [14] to define a new subclass of bi-univalent functions based on fractional derivative operator. Definition 1.1. Let the function 𝑓 (𝑧) be analytic in a simply connected region of the 𝑧− plane containing the origin. The fractional integral of 𝑓 of order 𝜇(𝜇 > 0) is defined by ∫𝑧 1 𝑓 (𝜉) 𝐷𝑧−𝜇 𝑓 (𝑧) = 𝑑𝜉 (3) Γ(𝜇) (𝑧 − 𝜉)1−𝜇 0

1−𝜇

where the multiplicity of (𝑧 − 𝜉) when 𝑧 − 𝜉 > 0.

is removed by requiring 𝑙𝑜𝑔(𝑧 − 𝜉) to be real

Definition 1.2. The fractional derivatives of order 𝜇(0 ≤ 𝜇 < 1) is defined for a function 𝑓 (𝑧) by ∫𝑧 1 𝑑 𝑓 (𝜉) 𝜇 𝐷𝑧 𝑓 (𝑧) = 𝑑𝜉 (4) Γ(1 − 𝜇) 𝑑𝑧 (𝑧 − 𝜉)𝜇 0

where the function 𝑓 (𝑧) is constrained and the multiplicity of the function (𝑧 − 𝜉)−𝜇 is removed as in Definition 1.1. Definition 1.3. Under the hypothesis of Definition 1.2, the fractional derivative of order 𝑛 + 𝜇 is defined by 𝐷𝑧𝑛+𝜇 𝑓 (𝑧) =

𝑑𝑛 𝜇 𝐷 𝑓 (𝑧), (0 ≤ 𝜇 < 1, 𝑛 ∈ ℕ0 ). 𝑑𝑧 𝑛 𝑧

(5)

With the aid of the above definitions and their known extensions involving fractional derivative and fractional integrals, Srivastava and Owa [25] introduced

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the operator Ω𝜇 : 𝒜 → 𝒜 given by Ω𝜇 𝑓 (𝑧) = Γ(2 − 𝜇)𝑧 𝜇 𝐷𝑧𝜇 𝑓 (𝑧) = 𝑧 +

∞ ∑ Γ(𝑛 + 1)Γ(2 − 𝜇) 𝑎𝑛 𝑧 𝑛 Γ(𝑛 + 1 − 𝜇) 𝑛=2

where 𝛿 ∈ ℝ(𝛿 ∕= 2, 3, 4, ..). It is of interest to note that Ω0 𝑓 (𝑧) = 𝑓 (𝑧) and Ω1 𝑓 (𝑧) = 𝑧𝑓 ′ (𝑧), further various choices 𝜇, (𝜇 ∈ ℝ) the operator Ω𝜇 itself includes Sal¨ 𝑎gean derivative operator and Libera integral operator (see [12, 13, 19, 25]). Now, we look at the definition of the fractional-order derivative of 𝑓 due to Caputo’s [2], which shall be used throughout the paper: 1 𝐷 𝑓 (𝑡) = Γ(𝑛 − 𝛼) 𝛼

∫𝑡

𝑓 (𝑛) (𝜏 ) 𝑑𝜏 (𝑡 − 𝜏 )𝛼+1−𝑛

(6)

𝑎

where 𝑛 − 1 < ℜ(𝛼) ≤ 𝑛, 𝑛 ∈ ℕ and the parameter 𝛼 is allowed to be real or even complex, 𝑎 is the initial value of the function 𝑓. We recall the following Caputo’s fractional operator defined in [12, 13, 19]. Definition 1.4. For functions 𝑓 ∈ 𝒜, we have ∫ Γ(2 + 𝜂 − 𝜇) 𝜇−𝜂 𝑧 Ω𝜂 𝑓 (𝑡) 𝒥𝜇𝜂 𝑓 (𝑧) = 𝑧 𝑑𝑡 𝜇+1−𝜂 Γ(𝜂 − 𝜇) 0 (𝑧 − 𝑡) ∞ ∑ (Γ(𝑛 + 1))2 Γ(2 + 𝜂 − 𝜇)Γ(2 − 𝜂) = 𝑧+ 𝑎𝑛 𝑧 𝑛 Γ(𝑛 + 𝜂 − 𝜇 + 1)Γ(𝑛 − 𝜂 + 1) 𝑛=2

(7)

where 𝜂(real number) 𝜂 − 1 < 𝜇 < 𝜂 < 2 and 𝑧 ∈ 𝕌. For the sake of brevity, we let (Γ(𝑛 + 1))2 Γ(2 + 𝜂 − 𝜇)Γ(2 − 𝜂) , Γ(𝑛 + 𝜂 − 𝜇 + 1)Γ(𝑛 − 𝜂 + 1) 4Γ(2 + 𝜂 − 𝜇)Γ(2 − 𝜂) 𝐶2 (𝜂, 𝜇) = Γ(3 + 𝜂 − 𝜇)Γ(3 − 𝜂)

𝐶𝑛 (𝜂, 𝜇) =

unless otherwise stated. Further, note that 𝒥00 𝑓 (𝑧) = 𝑓 (𝑧) and 𝒥11 𝑓 (𝑧) = 𝑧𝑓 ′ (𝑧).

(8) (9)

Coefficient Estimate of Bi-univalent Functions

437

Recently, especially after its revival by Srivastava et al. [22], there has been continuous interest in the study of the bi-univalent function class Σ leading to non-sharp coefficient estimates on the first two Taylor-Maclaurin coefficients ∣𝑎2 ∣ and ∣𝑎3 ∣. Many researchers (see, for example, [1, 3, 4, 5, 6, 8, 15, 24, 21, 28, 27, 29]) have recently introduced and investigated several interesting subclasses of the bi-univalent function class Σ and they have found non-sharp estimates on the corresponding first two Taylor-Maclaurin coefficients ∣𝑎2 ∣ and ∣𝑎3 ∣. Thus, motivated by the recent work of Goyal [5] we introduce a new subclass of biunivalent function class Σ by making use of the operator 𝒥𝜇𝜂 . For functions in this new subclass of Σ we obtain estimates of the coefficients ∣𝑎2 ∣ and ∣𝑎3 ∣. Several closely-related function classes are also considered and relevant connections to earlier known results are pointed out. Now we define a new subclass of Σ𝜂𝜇 (𝛽, 𝜆, ℎ) of the function class Σ. Definition 1.5. Let ℎ : 𝕌 → ℂ be a convex univalent function in 𝕌 such that ( ) ℎ(0) = 1 and ℜ ℎ(𝑧) > 0, (𝑧 ∈ 𝕌).

Suppose also that the function ℎ(𝑧) is given by ℎ(𝑧) = 1 +

∞ ∑

𝐵𝑛 𝑧 𝑛 , (𝑧 ∈ 𝕌).

(10)

𝑛=1

A function 𝑓 ∈ Σ given by (1) is said to be in the class Σ𝜂𝜇 (𝛽, 𝜆, ℎ), if it satisfies the following conditions:

and

) ( 𝒥𝜇𝜂 𝑓 (𝑧) ′ 𝑒𝑖𝛽 (1 − 𝜆) + 𝜆(𝒥𝜇𝜂 𝑓 (𝑧)) ≺ ℎ(𝑧)𝑐𝑜𝑠𝛽 + 𝑖𝑠𝑖𝑛𝛽 𝑧

(11)

( ) 𝒥𝜇𝜂 𝑔(𝑤) ′ 𝜂 𝑒 (1 − 𝜆) + 𝜆(𝒥𝜇 𝑔(𝑤)) ≺ ℎ(𝑤)𝑐𝑜𝑠𝛽 + 𝑖𝑠𝑖𝑛𝛽 𝑤

(12)

𝑖𝛽

𝜋 where 𝛽 ∈ ( −𝜋 2 , 2 ), 0 ≤ 𝜆 ≤ 1, the function 𝑔 is given by (2) and 𝑧, 𝑤 ∈ 𝕌.

By specializing the values of 𝜆, 𝜇 and 𝜂 we can further define new subclasses of Σ as illustrated below: Example 1.6. Let ℎ : 𝕌 → ℂ be a convex univalent function in 𝕌 such that ℎ(0) = 1 and ℜ(ℎ(𝑧)) > 0, 𝑧 ∈ 𝕌. Suppose also that the function ℎ(𝑧) is given by (10). A function 𝑓 ∈ Σ given by (1) is said to be in the class Σ𝜂𝜇 (𝛽, 0, ℎ) ≡ 𝒫Σ𝜇,𝜂 (𝛽, ℎ), if it satisfies the following conditions: 𝑒

𝑖𝛽

(

𝒥𝜇𝜂 𝑓 (𝑧) 𝑧

)

≺ ℎ(𝑧)𝑐𝑜𝑠𝛽 + 𝑖𝑠𝑖𝑛𝛽

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G. Murugusundaramoorthy and K. Thilagavathi

and 𝑒

𝑖𝛽

(

𝒥𝜇𝜂 𝑔(𝑤) 𝑤

)

≺ ℎ(𝑤)𝑐𝑜𝑠𝛽 + 𝑖𝑠𝑖𝑛𝛽

𝜋 where 𝛽 ∈ ( −𝜋 2 , 2 ), the function 𝑔 is given by (2) and 𝑧, 𝑤 ∈ 𝕌.

Example 1.7. Let ℎ : 𝕌 → ℂ be a convex univalent function in 𝕌 such that ℎ(0) = 1 and ℜ(ℎ(𝑧)) > 0, 𝑧 ∈ 𝕌. Suppose also that the function ℎ(𝑧) is given by (10). A function 𝑓 ∈ Σ given by (1) is said to be in the class Σ𝜂𝜇 (𝛽, 1, ℎ) ≡ 𝒬𝜇,𝜂 Σ (𝛽, ℎ), if it satisfies the following conditions: ( )′ 𝑒𝑖𝛽 𝒥𝜇𝜂 𝑓 (𝑧) ≺ ℎ(𝑧)𝑐𝑜𝑠𝛽 + 𝑖𝑠𝑖𝑛𝛽, and

( )′ 𝑒𝑖𝛽 𝒥𝜇𝜂 𝑔(𝑤) ≺ ℎ(𝑤)𝑐𝑜𝑠𝛽 + 𝑖𝑠𝑖𝑛𝛽

𝜋 where 𝛽 ∈ ( −𝜋 2 , 2 ), the function 𝑔 is given by (2) and 𝑧, 𝑤 ∈ 𝕌.

Example 1.8. Let ℎ : 𝕌 → ℂ be a convex univalent function in 𝕌 such that ℎ(0) = 1 and ℜ(ℎ(𝑧)) > 0, 𝑧 ∈ 𝕌. Suppose also that the function ℎ(𝑧) is given by (10). A function 𝑓 ∈ Σ given by (1) is said to be in the class Σ00 (𝛽, 𝜆, ℎ) ≡ ℱΣ (𝛽, 𝜆, ℎ), if it satisfies the following conditions: ) ( 𝑓 (𝑧) + 𝜆𝑓 ′ (𝑧) ≺ ℎ(𝑧)𝑐𝑜𝑠𝛽 + 𝑖𝑠𝑖𝑛𝛽 𝑒𝑖𝛽 (1 − 𝜆) 𝑧 and

( ) 𝑔(𝑤) 𝑒𝑖𝛽 (1 − 𝜆) + 𝜆𝑔 ′ (𝑤)) ≺ ℎ(𝑤)𝑐𝑜𝑠𝛽 + 𝑖𝑠𝑖𝑛𝛽 𝑤

𝜋 where 𝛽 ∈ ( −𝜋 2 , 2 ), 0 ≤ 𝜆 ≤ 1, the function 𝑔 is given by (2) and 𝑧, 𝑤 ∈ 𝕌.

Example 1.9. Let ℎ : 𝕌 → ℂ be a convex univalent function in 𝕌 such that ℎ(0) = 1 and ℜ(ℎ(𝑧)) > 0, 𝑧 ∈ 𝕌. Suppose also that the function ℎ(𝑧) is given by (10). A function 𝑓 ∈ Σ given by (1) is said to be in the class Σ11 (𝛽, 𝜆, ℎ) ≡ 𝒦Σ (𝛽, 𝜆, ℎ), if it satisfies the following conditions: 𝑒𝑖𝛽 (𝑓 ′ (𝑧) + 𝜆𝑧𝑓 ′′ (𝑧)) ≺ ℎ(𝑧)𝑐𝑜𝑠𝛽 + 𝑖𝑠𝑖𝑛𝛽 and 𝑒𝑖𝛽 (𝑔 ′ (𝑤) + 𝜆𝑤𝑔 ′′ (𝑤)) ≺ ℎ(𝑤)𝑐𝑜𝑠𝛽 + 𝑖𝑠𝑖𝑛𝛽 𝜋 where 𝛽 ∈ ( −𝜋 2 , 2 ), 0 ≤ 𝜆 ≤ 1, the function 𝑔 is given by (2) and 𝑧, 𝑤 ∈ 𝕌. 1+𝐴𝑧 Remark 1.10. If we set ℎ(𝑧) = 1+𝐵𝑧 , −1 ≤ 𝐵 < 𝐴 ≤ 1, then the class 𝜂 𝜂 Σ𝜇 (𝛽, 𝜆, ℎ) ≡ Σ𝜇 (𝛽, 𝜆, 𝐴, 𝐵) denotes the class of functions 𝑓 ∈ Σ, satisfying the following conditions: ( ) 𝒥𝜇𝜂 𝑓 (𝑧) 1 + 𝐴𝑧 𝑖𝛽 𝜂 ′ 𝑒 (1 − 𝜆) + 𝜆(𝒥𝜇 𝑓 (𝑧)) ≺ 𝑐𝑜𝑠𝛽 + 𝑖𝑠𝑖𝑛𝛽 (13) 𝑧 1 + 𝐵𝑧

Coefficient Estimate of Bi-univalent Functions

and

( ) 𝒥𝜇𝜂 𝑔(𝑤) 1 + 𝐴𝑤 𝜂 ′ (1 − 𝜆) + 𝜆(𝒥𝜇 𝑔(𝑤)) ≺ 𝑐𝑜𝑠𝛽 + 𝑖𝑠𝑖𝑛𝛽 𝑒 𝑤 1 + 𝐵𝑤 𝑖𝛽

439

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𝜋 where 𝛽 ∈ ( −𝜋 2 , 2 ), 0 ≤ 𝜆 ≤ 1, the function 𝑔 is given by (2) and 𝑧, 𝑤 ∈ 𝕌.

, 0 ≤ 𝛼 < 1 then the class Σ𝜂𝜇 (𝛽, 𝜆, ℎ) Remark 1.11. If we set ℎ(𝑧) = 1+(1−2𝛼)𝑧 1−𝑧 𝜂 ≡ Σ𝜇 (𝛽, 𝜆, 𝛼) denotes the class of functions 𝑓 ∈ Σ, such that ( [ ]) 𝒥𝜇𝜂 𝑓 (𝑧) 𝑖𝛽 𝜂 ′ ℜ 𝑒 (1 − 𝜆) + 𝜆(𝒥𝜇 𝑓 (𝑧)) > 𝛼𝑐𝑜𝑠𝛽 (15) 𝑧 and

( [ ]) 𝒥𝜇𝜂 𝑔(𝑤) ℜ 𝑒𝑖𝛽 (1 − 𝜆) + 𝜆(𝒥𝜇𝜂 𝑔(𝑤))′ > 𝛼𝑐𝑜𝑠𝛽 𝑤

(16)

𝜋 where 𝛽 ∈ ( −𝜋 2 , 2 ), 0 ≤ 𝜆 ≤ 1, the function 𝑔 is given by (2) and 𝑧, 𝑤 ∈ 𝕌.

Remark 1.12. Suitably specializing the parameters 𝜆, 𝜂 and 𝜇 in the Remarks 1.10 and 1.11, as illustrated in Examples 1.6 to 1.9, one can define various subclasses of Σ𝜂𝜇 (𝛽, 𝜆, 𝐴, 𝐵) and Σ𝜂𝜇 (𝛽, 𝜆, 𝛼) respectively.

2. Coefficient Estimates In order to prove our main result for the functions class 𝑓 ∈ Σ𝜂𝜇 (𝛽, 𝜆, ℎ) we first recall the following lemma. Lemma 2.1. [18] Let the function 𝜑(𝑧) given by 𝜑(𝑧) =

∞ ∑

𝐶𝑛 𝑧 𝑛 , (𝑧 ∈ 𝕌)

𝑛=1

be convex in 𝕌. Suppose that the function ℎ(𝑧) given by ℎ(𝑧) =

∞ ∑

ℎ𝑛 𝑧 𝑛 ,

𝑛=1

is holomorphic in 𝕌. If ℎ(𝑧) ≺ 𝜑(𝑧), (𝑧 ∈ 𝕌) then ∣ℎ𝑛 ∣ ≤ ∣𝐶1 ∣, (𝑛 ∈ ℕ). Lemma 2.2. [16] If a function 𝑝 ∈ 𝒫, is given by 𝑝(𝑧) = 1 + 𝑝1 𝑧 + 𝑝2 𝑧 2 + ⋅ ⋅ ⋅ (𝑧 ∈ 𝕌)

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G. Murugusundaramoorthy and K. Thilagavathi

then ∣𝑝𝑘 ∣ ≤ 2 for each 𝑘, where 𝒫 is the family of all functions 𝑝 analytic in 𝕌 for which 𝑝(0) = 1 and ℜ (𝑝(𝑧)) > 0.

Theorem 2.3. Let the function 𝑓 given by (1) be in the class Σ𝜂𝜇 (𝛽, 𝜆, ℎ). Suppose also that 𝐵1 is given as in the Taylor-Maclaurin expansion (10) of the function ℎ(𝑧). Then √ ∣𝑎2 ∣ ≤

∣𝐵1 ∣𝑐𝑜𝑠𝛽 𝐶3 (𝜂, 𝜇)(1 + 2𝜆)

and ∣𝐵1 ∣𝑐𝑜𝑠𝛽 + ∣𝑎3 ∣ ≤ 𝐶3 (𝜂, 𝜇)(1 + 2𝜆)

(

∣𝐵1 ∣𝑐𝑜𝑠𝛽 𝐶2 (𝜂, 𝜇)(1 + 𝜆)

(17)

)2

(18)

𝜋 where 𝛽 ∈ ( −𝜋 2 , 2 ) and 0 ≤ 𝜆 ≤ 1.

Proof. From (11) and (12), we have ) ( 𝒥𝜇𝜂 𝑓 (𝑧) ′ + 𝜆(𝒥𝜇𝜂 𝑓 (𝑧)) = 𝑝(𝑧)𝑐𝑜𝑠𝛽 + 𝑖𝑠𝑖𝑛𝛽, (𝑧 ∈ 𝕌) 𝑒𝑖𝛽 (1 − 𝜆) 𝑧

(19)

and ( ) 𝒥𝜇𝜂 𝑔(𝑤) ′ 𝜂 𝑒 (1 − 𝜆) + 𝜆(𝒥𝜇 𝑔(𝑤)) = 𝑞(𝑤)𝑐𝑜𝑠𝛽 + 𝑖𝑠𝑖𝑛𝛽, (𝑤 ∈ 𝕌) 𝑤 𝑖𝛽

(20)

where 𝑝(𝑧) ≺ ℎ(𝑧), (𝑧 ∈ 𝕌) and 𝑞(𝑤) ≺ ℎ(𝑤), (𝑤 ∈ 𝕌) have the following forms: 𝑝(𝑧) = 1 + 𝑝1 𝑧 + 𝑝2 𝑧 2 + 𝑝3 𝑧 3 + ......, (𝑧 ∈ 𝕌)

(21)

𝑞(𝑤) = 1 + 𝑞1 𝑤 + 𝑞2 𝑤2 + 𝑞3 𝑤3 + ......, (𝑤 ∈ 𝕌).

(22)

and Now, equating the coefficients in (19) and (20), we get 𝑒𝑖𝛽 𝐶2 (𝜂, 𝜇)(1 + 𝜆)𝑎2 = 𝑝1 𝑐𝑜𝑠𝛽

(23)

𝑒𝑖𝛽 𝐶3 (𝜂, 𝜇)(1 + 2𝜆)𝑎3 = 𝑝2 𝑐𝑜𝑠𝛽

(24)

−𝑒𝑖𝛽 𝐶2 (𝜂, 𝜇)(1 + 𝜆)𝑎2 = 𝑞1 𝑐𝑜𝑠𝛽

(25)

𝑒𝑖𝛽 𝐶3 (𝜂, 𝜇)(1 + 2𝜆)(2𝑎22 − 𝑎3 ) = 𝑞2 𝑐𝑜𝑠𝛽.

(26)

and From (23) and (25), it follows that 𝑝1 = −𝑞1

(27)

Coefficient Estimate of Bi-univalent Functions

441

and (𝑝21 + 𝑞12 )𝑐𝑜𝑠2 𝛽 = 𝑒2𝑖𝛽 2[𝐶2 (𝜂, 𝜇)]2 (1 + 𝜆)2 𝑎22 . Hence we have 𝑎22 =

(𝑝21 + 𝑞12 )𝑐𝑜𝑠2 𝛽(𝑒−2𝑖𝛽 ) . 2[𝐶2 (𝜂, 𝜇)(1 + 𝜆)]2

(28)

(29)

Adding (24)and (26), it follows that 𝑎22 =

(𝑝2 + 𝑞2 ) 𝑐𝑜𝑠𝛽𝑒−𝑖𝛽 . 2𝐶3 (𝜂, 𝜇)(1 + 2𝜆)

(30)

Suppose 𝑝(𝑧), 𝑞(𝑤) ⊂ ℎ(𝕌), by applying Lemma 2.1 for the coefficients 𝑝2 and 𝑞2 we get 𝑘 𝑝 (0) ≤ ∣𝐵1 ∣, (𝑘 ∈ ℕ) ∣𝑝𝑘 ∣ = (31) 𝑘! and

𝑘 𝑞 (0) ≤ ∣𝐵1 ∣, (𝑘 ∈ ℕ). ∣𝑞𝑘 ∣ = 𝑘!

(32)

Using above equations (27),(31) and (32), we have ∣𝑎2 ∣2 ≤

(∣𝐵1 ∣)𝑐𝑜𝑠𝛽 (∣𝑞2 ∣ + ∣𝑝2 ∣)𝑐𝑜𝑠𝛽 ≤ 2𝐶3 (𝜂, 𝜇)(1 + 2𝜆) 𝐶3 (𝜂, 𝜇)(1 + 2𝜆)

(33)

which gives the estimate on ∣𝑎2 ∣ as asserted in (17). Subtracting from (24) and (26),we get 𝑎3 − 𝑎22 =

(𝑝2 − 𝑞2 )𝑒−𝑖𝛽 𝑐𝑜𝑠𝛽 . 2(𝐶3 (𝜂, 𝜇))(1 + 2𝜆)

(34)

Substituting value of 𝑎22 from (29) in (34) we get 𝑎3 =

(𝑝2 − 𝑞2 )𝑒−𝑖𝛽 𝑐𝑜𝑠𝛽 (𝑝2 + 𝑞12 )𝑐𝑜𝑠2 𝛽𝑒−2𝑖𝛽 + 1 . 2𝐶3 (𝜂, 𝜇)(1 + 2𝜆) 2[𝐶2 (𝜂, 𝜇)(1 + 𝜆)]2

(35)

Applying Lemma 2.1 once again for the coefficients 𝑝1 , 𝑝2 , 𝑞1 and 𝑞2 , we get ∣𝐵1 ∣𝑐𝑜𝑠𝛽 ∣𝑎3 ∣ ≤ + 𝐶3 (𝜂, 𝜇)(1 + 2𝜆)

(

∣𝐵1 ∣𝑐𝑜𝑠𝛽 𝐶2 (𝜂, 𝜇)(1 + 𝜆)

)2

(36)

which gives the estimate on ∣𝑎3 ∣ as asserted in (18). 1+𝐴𝑧 By taking ℎ(𝑧) = 1+𝐵𝑧 , −1 ≤ 𝐵 < 𝐴 ≤ 1 we state the following (analogous estimates asserted in Theorem 2.3) corollary for the function classes defined in Remark 1.10 without proof.

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Corollary 2.4. Let the function 𝑓 given by (1) be in the class Σ𝜂𝜇 (𝛽, 𝜆, 𝐴, 𝐵). Then √ (𝐴 − 𝐵)𝑐𝑜𝑠𝛽 ∣𝑎2 ∣ ≤ (37) 𝐶3 (𝜂, 𝜇)(1 + 2𝜆) and ∣𝑎3 ∣ ≤

(𝐴 − 𝐵)𝑐𝑜𝑠𝛽 + 𝐶3 (𝜂, 𝜇)(1 + 2𝜆)

𝜋 where 𝛽 ∈ ( −𝜋 2 , 2 ) and 0 ≤ 𝜆 ≤ 1.

(

(𝐴 − 𝐵)𝑐𝑜𝑠𝛽 𝐶2 (𝜂, 𝜇)(1 + 𝜆)

)2

(38)

By taking ℎ(𝑧) = 1+(1−2𝛼)𝑧 , ( 0 ≤ 𝛼 < 1) we state the following(analogous 1−𝑧 estimates asserted in Theorem 2.3) corollary for the function classes defined in Remark 1.11) without proof. Corollary 2.5. Let the function 𝑓 given by (1) be in the class Σ𝜂𝜇 (𝛽, 𝜆, 𝛼). Then √ 2(1 − 𝛼)𝑐𝑜𝑠𝛽 ∣𝑎2 ∣ ≤ (39) 𝐶3 (𝜂, 𝜇)(1 + 2𝜆) and ∣𝑎3 ∣ ≤

2(1 − 𝛼)𝑐𝑜𝑠𝛽 + 𝐶3 (𝜂, 𝜇)(1 + 2𝜆)

𝜋 where 𝛽 ∈ ( −𝜋 2 , 2 ) and 0 ≤ 𝜆 ≤ 1.

(

2(1 − 𝛼)𝑐𝑜𝑠𝛽 𝐶2 (𝜂, 𝜇)(1 + 𝜆)

)2

(40)

Concluding Remarks: Specializing the parameters 𝜆, 𝜇 and 𝜂, various other interesting estimates for the subclasses defined in Examples 1.6 to 1.9 (which are asserted in Theorem 2.3) can be derived similarly and so we omit the details. Further by choosing 𝛽 = 0 we can state the results analogous to the results studied in [4, 22]. Acknowledgement. The authors thank the referees for their valuable suggestions. We the authors record our sincere thanks to Prof. H.M. Srivastava for his insightful suggestions.

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