Introduction Let Adenote the class of the functions f of the form f(z) =z

COEFFICIENT BOUNDS FOR NEW SUBCLASSES OF BI-UNIVALENT FUNCTIONS USING HADAMARD PRODUCT

A. G. Alamoush and M. Darus

Abstract. The aim of the present paper is to introduce a new subclass of bi-univalent functions defined in the open unit disc using Hadamard product. We obtain estimates on the coefficients |a₂| and |a₃| for functions of this class. Some results related to this work will also be pointed out.

2000Mathematics Subject Classification: 30C45.

Keywords: Analytic and univalent functions, Bi-univalent functions, Starlike and convex functions, Coefficients bounds.

1. Introduction Let Adenote the class of the functions f of the form

f(z) =z+

∞

X

n=2

a_nzⁿ (1)

which are analytic in the open unit disc U = {z ∈ C : |z| < 1} and satisfy the normalization condition f(0) =f⁰(0) = 0. Let S be the subclass of A consisting of functions of the form (1) which are also univalent in U. For n ∈N0, we introduce the subclassQ(n, δ, β, λ) ofS of functionsf of the form (1), satisfying the condition

Ren _(1−λ)Dk

n,δf(z)+λD^k+1

n,δ f(z) z

o

> β, z ∈U, (2)

where D_n,δ^k is the differential operator given by Hadamard product between Salagean and Ruscheweyh operators, such as

D_n,δ^k f(z) =z+P∞

n=2C(δ, n)n^kanzⁿ.

Fork=δ = 0, it reduces to the classQ_λ(β) studied by Ding et al. [3], (see also [4-7]).

Now by having

f⁻¹f(z) =z, (z∈U), and

f⁻¹f(w) =w, (|w|< r0, f(z)≥ ¹₄ )

wheref⁻¹(w) =w−a2w²+ (2a²₂−a3)w³−(5a²₂−5a2a3+a4)w⁴+... , we say that a function f(z)∈A is bi-univalent inU if bothf(z) andf⁻¹(z) are univalent inU. Let Σ denote the class of bi-univalent functions in U given by (1). For a brief history and interesting examples in the class Σ, see [8]. In fact, Brannan and Taha [9] (see also [11]) introduced certain subclasses of the bi-univalent functions similar to the familiar subclasses S^∗(α) andK(α) of starlike and convex functions of order α(0≤α < 1), respectively (see [10]). Following the same manner of Brannan and Taha [9] (see also [11]), a function f ∈ A is in the class of strongly bi-Starlike functions of order α(0< α ≤1) if each of the following conditions is satisfied: For f ∈Σ,

arg

zf⁰(z) f(z)

< ^πα₂ ,α(0< α≤1, z ∈U), and

arg

wg⁰(w) g(w)

< ^πα₂ ,α(0< α≤1, w∈U),

where g is the extension of f⁻¹(z) to U. Similarly, a function f ∈A is in the class KΣ(α) of strongly bi-convex functions of orderα if each of the following conditions are satisfied: For f ∈Σ,

arg

1 +^zf

00(z) f⁰(z)

< ^πα₂ ,α(0< α≤1, z∈U), and

arg

1 +^wg

00(w) g⁰(w)

< ^πα₂ ,α(0< α≤1, w∈U),

where g is the extension of to U. The classes S^∗_Σ(α) and K_Σ(α) of bi-starlike func- tions of order α and bi-convex functions of orderα, corresponding (respectively) to the classes of S^∗(α) and K(α) were also introduced analogously. For each of the classesS_Σ^∗(α) andK_Σ(α), it was noted that the estimates obtained for the first two coefficients |a₂|and|a₃|are not sharp (for details, see [9,11]).

The object of the paper is to introduce two new subclasses of the function class Σ and to find estimates on the coefficients |a₂| and |a₃|using the same techniques given earlier by Srivastava et al. [8], Frasin and Aouf [12], and Porwal and Darus [2]. In order to prove our main results, we need the following lemma due to [15].

Lemma 1. If h∈p then|c_k|<1,for each k, where p is the family of all functions h analytic in U for which Re{h(z)}>0, then

h(z) = 1 +c1z+c2z²+c3z³+... , z∈U.

2. Coefficient bounds for the function class Q_Σ(n, δ, α, λ)

Definition 1. A functionf(z) given by (1) is said to be in the class QΣ(n, δ, α, λ) if the following conditions are satisfied: For f ∈Σ,

arg(1−λ)D^k_n,δf(z) +λD_n,δ^k+1f(z) z

< πα

2 , α(0< α≤1, λ≥1, z∈U), (3) and

arg(1−λ)D^k_n,δg(w) +λD_n,δ^k+1g(w) w

< πα

2 , α(0< α≤1, λ≥1, w∈U), (4) where the function g is given by

g(w) =w−a2w²+ (2a²₂−a3)w³−(5a²₂−5a2a3+a4)w⁴+... . (5) We note that fork =δ = 0, λ = 1, the class Q_Σ(n, δ, α, λ) reduces to the class H_Σ^α introduced and studied by Srivastava et al [8], fork=δ = 0,the class reduces to Q_Σ(α, λ) introduced and studied by Frasin and Aouf [12]. Also for δ= 0, the class Q_Σ(n, δ, α, λ) reduces to Q_Σ(n, α, λ) studied by Porwal and Darus [2]. We begin by finding the estimates of the coefficients for functions in the class QΣ(n, δ, α, λ).

Theorem 2. Let the function f(z) given by (1) be in the class Q_Σ(n, δ, α, λ), n∈N₀,0≤β <1, λ≥1. Then

|a₂| ≤4α

Γ(δ+ 1) Γ(δ+ 2)

"

1

p4^k(1 +λ)²+α[2.3^k(1 +λ)−4^k(1 +λ)²]

#

(6) and

|a₃| ≤12αΓ(δ+ 1) Γ(δ+ 3)

1

(1−λ)3^k+λ3^k(1 +λ) + 2α

[(1−λ)2^k+λ2^k+1]²

(7) Proof. From (3) and (4), we can write

(1−λ)D_n,δ^k f(z) +λD_n,δ^k+1f(z)

z = [p(z)]^α, (8)

and

(1−λ)D^k_n,δg(w) +λD_n,δ^k+1g(w)

w = [q(w)]^α, (9)

respectively, where p(z) and q(w) are inp and have the form

p(z) = 1 +p₁z+p₂z²+p₃z³+... , (10) and

q(w) = 1 +p₁w+q₂w²+q₃w³+... . (11) Now, equating the coefficients in (8) and (9), we obtain

[(1−λ)2^k+λ2^k+1]C(δ,2)a2 =αp1, (12) [(1−λ)3^k+λ3^k+1]C(δ,3)a₃ = 1

2[2αp₂+α(α−1)p²₁], (13)

−[(1−λ)2^k+λ2^k+1]C(δ,2)a2 =αq1, (14) [(1−λ)3^k+λ3^k+1](2[C(δ,2)]²a²₂−C(δ,3)a₃) = 1

2[2αq₂+α(α−1)q₁²]. (15) From (12) and (14), we obtain

p₁ =−q₁ (16)

and

2[(1−λ)2^k+λ2^k+1]²[C(δ,2)]²a²₂=α²(p²₁+q₁²). (17)

2[(1−λ)3^k+λ3^k+1][C(δ,2)]²a²₂ =α(p2+q2) +¹₂[α(α−1)(p²₁+q²₁)]

=α(p2+q2) +^α(α−1)₂ .^{2[(1−λ)2k+λ2}_α^k+12 ^{]2[C(δ,2)]2a22} . Therefore we have

a²₂= _[4k(1+λ)²+α[2.3^kα(1+λ)]−4^2(p2+q2)k(1+λ)²]]C[(δ,2)]².

Applying Lemma 1 for the coefficientsp₂ and q₂, we immediately have

|a₂| ≤4α

Γ(δ+1) Γ(δ+2)

√ 1

4^k(1+λ)²+α[2.3^k(1+λ)−4^k(1+λ)²]

. This gives the bound as asserted in (6).

Next, in order to find the bound on |a₃|, we subtract (13) from (15) and obtain 2[(1−λ)3^k+λ3^k+1](C(δ,3)a3−C[(δ,2)]²a²₂)

=¹₂(2α(p₂−q₂) +α(α−1)(p²₁−q₁²)), a₃ = _2[(1−λ)3^α(p_k+λ3^2−q_k+1²⁾_](Cδ,3) +_2[(1−λ)2^α_k²_+λ2^(p21+q_k+1¹²⁾_]2(Cδ,3), a3= _2[(1−λ)3^6α(p2k^−q+λ3^{2)Γ(δ+1)k+1}]Γ(δ+3)+ _2[(1−λ)2^6Γ(δ+1)(αk+λ2^2k+1)(p21]^2+qΓ(δ+3)¹²⁾ . Applying Lemma 1 for the coefficientsp2 and q2, we immediately have

|a₃| ≤ _[(1−λ)3^12αΓ(δ+1)_{k+λ3k+1]Γ(δ+3)} +_[(1−λ)2^24Γ(δ+1)αk+λ2^k+1]²²Γ(δ+3), i.e.

|a₃| ≤12α^Γ(δ+1)_Γ(δ+3)

h 1

(1−λ)3^k+λ3^k(1+λ) +_[(1−λ)2k^2α+λ2^k+1]²

i . This completes the proof of Theorem 2.

Puttingλ= 1, k=δ= 0,in Theorem 2, we have

Corollary 3. Let f(z) given by (1) be in the class H_Σ^α(0< α≤1). Then

|a₂| ≤αq

2 2+α

and

|a₃| ≤ ^α(2+3α)₃ .

3. Coefficient bounds for the function class H_Σ(n, δ, β, λ)

Definition 2. A functionf(z) given by (1) is said to be in the classH_Σ(n, δ, β, λ) if the following conditions are satisfied:

Re

((1−λ)D_n,δ^k f(z) +λD^k+1_n,δ f(z) z

)

> β, z ∈U, n∈N0,0≤β <1, λ≥1. (18) and

Re

((1−λ)D_n,δ^k g(w) +λD^k+1_n,δ g(w) w

)

> β, w∈U, n∈N₀,0≤β <1, λ≥1 (19) where the function g is defined by (5).

We note that fork = δ = 0, and λ= 1, H_Σ(n, δ, β, λ) the class reduced to the classes HΣ(β) studied by Srivastava et al.[8], and for k= δ = 0, the class reduced to the classes H_Σ(β, λ) studied by Frasin and Aouf [12].

Theorem 4. Let the function f(z) given by (1) be in the class HΣ(n, δ, β, λ), n∈N0,0≤β <1, λ≥1. Then

|a₂| ≤2

Γ(δ+ 1) Γ(δ+ 2)

s

2(1−β)

(1−λ)3^k+λ3^k+1 (20)

and

|a₃| ≤ 12(1−β)Γ(δ+ 1) Γ(δ+ 3)

2(1−β)

[(1−λ)2^k+λ2^k+1]² + 1

(1−λ)3^k+λ3^k+1

. (21) Proof. It follows from (18) and (19) that there existsp, q∈P such that

(1−λ)D^k_n,δf(z) +λD^k+1_n,δ f(z)

z =β+ (1−β)p(z), (22)

and

(1−λ)D_n,δ^k g(w) +λD^k+1_n,δ g(w)

w =β+ (1−β)q(w), (23)

wherep(z) andq(w) have the forms (10) and (11), respectively. Equating coefficients in (22) and (23) yields

[(1−λ)2^k+λ2^k+1]C(δ,2)a₂ = (1−β)p₁, (24)

−[(1−λ)2^k+λ2^k+1]C(δ,2)a2= (1−β)q1, (26) and

[(1−λ)3^k+λ3^k+1](2[C(δ,2)]²a²₂−C(δ,3)a₃) = (1−β)q₂. (27) From (24) and (26), we have

−p₁ =q₁ (28)

and

2[(1−λ)2^k+λ2^k+1]²C[(δ,2)]²a²₂ = (1−β)²(p²₁+q₁²). (29) Also, from (25) and (27), we find that

2[(1−λ)3^k+λ3^k+1]C[(δ,2)]²a²₂= (1−β)(p₂+q₂), (30)

|a²₂| ≤ (1−β)(|p₂|+|q₂|)

2[(1−λ)3^k+λ3^k+1]C[(δ,2)]², (31) i.e.

|a₂| ≤2

Γ(δ+ 1) Γ(δ+ 2)

s

2(1−β)

(1−λ)3^k+λ3^k+1. (32) which is the bound on|a₂|as given in (20).

Next, in order to find the bound on|a₃|by subtracting (27) from (25), we obtain 2C(δ,3)[(1−λ)3^k+λ3^k+1]a3=

2[(1−λ)3^k+λ3^k+1][C(δ,2)]²a²₂+ (1−β)(p₂−q₂) or, equivalently

a3= ^2[(1−λ)32C(δ,3)[(1−λ)3^{k+λ3k+1k][C(δ,2)]}+λ3^k+12]^a22 +2C(δ,3)[(1−λ)3^{(1−β)(p2−qk}+λ3^{2) k+1}]

Upon substituting the value of a²₂ from (29), we obtain a₃ = 3(1−β)²(p²₁+q²₁)Γ(δ+ 1)

[(1−λ)2^k+λ2^k+1]²Γ(δ+ 3) + 3(1−β)(p₂−q₂)Γ(δ+ 1)

[(1−λ)3^k+λ3^k+1]Γ(δ+ 3). (33) Applying Lemma 1 for the coefficientsp₁, p₂, q₁ and q₂ we obtain

|a₃| ≤ 12(1−β)Γ(δ+ 1) Γ(δ+ 3)

2(1−β)

[(1−λ)2^k+λ2^k+1]² + 1

(1−λ)3^k+λ3^k+1

(34) which is the bound on |a₃|as asserted in (21).

Puttingλ= 1, k=δ = 0,in Theorem 4, we have the following corollary.

Corollary 5. Let f z) given by (1) be in the classHΣ(n, δ, β, λ),(0≤β <1). Then

|a₂| ≤

r2(1−β)

3 (35)

and

|a₃| ≤ (1−β)(5−3β)

3 . (36)

Remark 1. If we put δ=k= 0, in Theorems 2 and 3, we obtain the corresponding results due to Frasin and Aouf [12].

Remark 2. If we put δ = 0, in Theorems 2 and 3, we obtain the corresponding results due to Porwal and Darus [2].

Remark 3. If we put δ = k = 0, λ = 1, in Theorems 2 and 3, we obtain the corresponding results due to Srivastava et al [8].

Remark 4. Similarly, just as stated in [2], it would be nice to find estimates for

|a_n|, n≥4 (not necessarily sharp) for the class of functions defined in this work.

Acknowledgement: The work here is partially supported by UKM’s grant: GUP- 2013-004.

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1Adnan G. Alamoush and²Maslina Darus

1,2 School of Mathematical Sciences Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi Selangor, malaysia

email: ¹adnan–[email protected], ²[email protected]