Coefficient bounds for some families of starlike and convex functions of complex

Coefficient bounds for some families of starlike and convex functions of complex

order

Sevtap S¨umer Eker, Bilal S¸eker , Mugur Acu

Abstract

2000 Mathematics Subject Classification: Primary 30C45, 30C50.

Key words and phrases: Dziok-Srivastava Operator; Coefficient bounds; Starlike functions of complex order; Convex functions of complex

order; Nonhomogeneous Cauchy-Euler differential equations.

1 Introduction and preliminaries

Let A denote the class of functions of the form f(z) =z+

X∞

k=2

a_kz^k, (1.1)

which are analytic in the open unit disk U={z ∈C:|z|<1}.

195

given by

g(z) = z+ X∞

k=2

bkz^k,

the Hadamard product(or convolution)f ∗g is defined (as usual) by (f ∗g) (z) = z+

X∞

k=2

a_kb_kz^k = (g∗f)(z) , z ∈U.

Forα_j ∈C (j = 1,2, ...q) andβ_j ∈C\Z⁻₀ ; Z⁻₀ ={0,−1,−2, ...} (j = 1,2, ...s) the generalized hypergeometric function_qF_s(α₁, ..., α_q;β₁, ..., β_s;z) is defined by

qF_s(α₁, ..., α_q;β₁, ..., β_s;z) = X∞

k=0

(α₁)_k...(α_q)_k (β₁)_k...(β_s)_k

z^k k!

(q≤s+ 1, q, s∈N₀ ={0,1,2, ...}=N∪ {0}).

Here, and in what follows, (κ)n denotes the Pochhammer symbol (or shifted factorial) defined, in terms of the Gamma function Γ, by

(κ)n= Γ(κ+n) Γ(κ) =





1 n = 0, κ6= 0

κ(κ+ 1)...(κ+n−1) n∈N For the function

h(α₁, ..., α_q;β₁, ..., β_s;z) = z_qF_s(α₁, ..., α_q;β₁, ..., β_s;z)

the Dziok-Srivastava linear operator [2] (see also [3])H_s^q(α₁, ..., α_q;β₁, ..., β_s;z) is defined by the following Hadamard product (or convolution) :

H_s^q(α₁, ..., α_q;β₁, ..., β_s;z)f(z) = h(α₁, ..., α_q;β₁, ..., β_s;z)∗f(z)

=z+ X∞

k=2

(α₁)_k−1...(α_q)_k−1 (β1)k−1...(βs)k−1

1

(k−1)!a_kz^k. For notational simplicity, we write

H_s^q(α₁, ..., α_q;β₁, ..., β_s;z)f(z) = (H_s^q[α₁]f) (z)

A function f(z) ∈ A is said to be in the class S^∗(b) if it satisfies the following inequality:

Re

· 1 + 1

b

µzf⁰(z) f(z) −1

¶¸

>0 (z∈U, b ∈C^∗ =C\ {0}) Furthermore, a functionf(z)∈ Ais said to be in the class C(b) if it also satisfies the following inequality:

Re

· 1 + 1

b

µzf⁰⁰(z) f⁰(z)

¶¸

>0 (z ∈U, b∈C^∗)

The function classesS^∗(b) and C(b) were considered earlier by Nasr and Aouf [4-6] and Wiatrowski [7], respectively.

Furthermore, a function f(z)∈ A is said to be in the class SC(b, λ, γ)

Re

· 1 + 1

b

µz[λzf⁰(z) + (1−λ)f(z)]⁰ λzf⁰(z) + (1−λ)f(z) −1

¶¸

> γ (1.2)

(f(z)∈ A; 0≤λ≤1 ; 0 ≤γ <1 ; b∈ C^∗; z ∈U).

Altınta¸s et al. [1].

Let SC^q,s_α,β(b, λ, γ) denote the subclass of A consisting of functions f(z) which satisfy the following condition:

Re

½ 1 + 1

b

µz[λz(H_s^q[α1]f)⁰(z) + (1−λ)(H_s^q[α1]f)(z)]⁰ λz(Hs^q[α₁]f)⁰(z) + (1−λ)(Hs^q[α₁]f)(z) −1

¶¾

> γ, (1.3)

(f(z)∈ A, 0≤λ≤1, 0≤γ <1, b∈C^∗, q≤s+ 1, q, s∈N₀, z ∈U).

For q = 2, s = 1, α1 = β1, α2 = 1, we obtain the class of SC(b, λ, γ).

Furthermore, for q = 2, s = 1, α₁ = β₁, α₂ = 1, γ = 0 and λ = 0 the class SC^q,s_α,β(b, λ, γ) is coincide the classS^∗(b) and forq = 2, s= 1, α1 =β1, α₂ = 1, γ = 0 andλ = 1 we obtain the class C(b).

The main object of the present investigation is to derive some coefficient bounds for functions in the subclass T_α,β^q,s(b, λ, γ;µ) of A which consists of functions f(z)∈ A satisfying the following nonhomogeneous Cauchy-Euler differential equation:

z²d²w

dz² + 2(1 +µ)zdw

dz +µ(1 +µ)w= (1 +µ)(2 +µ)g(z) (1.4)

¡w=f(z)∈ A, g(z)∈ SC^q,s_α,β(b, λ, γ), µ ∈R\(−∞,−1]¢ .

2 Main results

Theorem 1. Let the function f(z)∈ A be defined by (1.1). If the function f(z) is in the class SC^q,s_α,β(b, λ, γ), then

|a_k| ≤

k−2Y

j=0

[j+ 2|b|(1−γ)]

[1 +λ(k−1)]^(α_(β¹₁⁾₎^{k−1...(αq)k−1}

k−1···(βs)k−1

(k ∈N^∗ =N\{1}={2,3,4, ...}) (2.1) Proof. Let the functionf(z)∈ Abe given by (1.1) and let the function F(z) be defined by

F(z) =λz(H_s^q[α₁]f)⁰(z) + (1−λ)(H_s^q[α₁]f)(z), (f(z)∈ A, 0≤λ≤1).

Then from (1.3) and the definition ofF(z) above, it is easily seen that

Re

· 1 + 1

b

µzF⁰(z) F(z) −1

¶¸

> γ with

F(z) = z+ X∞

k=2

A_kz^k ∈ A

µ

A_k = [1 +λ(k−1)](α₁)_k−1· · ·(α_q)_k−1 (β₁)_k−1· · ·(β_s)_k−1

1

(k−1)!a_k, k ∈N^∗

¶

Thus, by setting

1 + ¹_b

³zF⁰(z) F(z) −1

´

−γ

1−γ =p(z)

zF⁰(z) = [1 +b(1−γ)(p(z)−1)]F(z), (2.2) we get

p(z) = 1 +p1z+p2z²+· · · (z ∈U). (2.3) Since

Re(p(z))>0, 0≤γ <1 ; b∈C^∗ we conclude that

|p_k| ≤2 (k ∈N).

We also find from (2.2) and (2.3) that

(k−1)Ak =b(1−γ) (p1Ak−1+p2Ak−2+· · ·+pk−1). In particular, for k = 2,3,4, we have

A2 =b(1−γ)p1 implies |A2| ≤2|b|(1−γ)

2A₃ =b(1−γ)(p₁A₂+p₂) implies |A₃| ≤ 2|b|(1−γ)[1 + 2|b|(1−γ)]

2!

and

3A4 =b(1−γ)(p1A3+p2A2+p3)

implies

|A₄| ≤ 2|b|(1−γ)[1 + 2|b|(1−γ)][2 + 2|b|(1−γ)]

3!

respectively. Using the principle of mathematical induction, we obtain,

|A_k| ≤

k−2Y

j=0

[j+ 2|b|(1−γ)]

(k−1)! (k ∈N^∗).

Moreover, by the relationship between the functions f(z) and F(z), it is clear that

Ak = [1 +λ(k−1)](α₁)_k−1· · ·(α_q)_k−1 (β₁)_k−1· · ·(β_s)_k−1

1

(k−1)!ak (k ∈N^∗).

and then we get

|ak| ≤

k−2Y

j=0

[j + 2|b|(1−γ)]

[1 +λ(k−1)]^(α_(β¹₁⁾₎^k−1_k−1^...(α_···(βs^q₎⁾_k−1^k−1.

By choosing suitable values of the admissible parametersb, λ, γ, α and β in Theorem 1 above, we deduce the following corollaries.

Corollary 1. (Altınta¸s et al. [1]) If a function f(z) ∈ A is in the class SC(b, λ, γ), then

|a_k| ≤

k−2Y

j=0

[j+ 2|b|(1−γ)]

(k−1)![1 +λ(k−1)] (k∈N^∗).

S^∗(b), then

|ak| ≤

k−2Y

j=0

[j + 2|b|]

(k−1)! (k∈N^∗).

Corollary 3. (Nasr and Aouf [4]) If a function f(z) ∈ A is in the class C(b), then

|ak| ≤

k−2Y

j=0

[j + 2|b|]

k! (k∈N^∗).

Theorem 2. Let the functionf(z)∈ A be defined by (1.1). If the function f(z) is in the class T_α,β^q,s(b, λ, γ;µ), then

|a_k| ≤

(1 +µ)(2 +µ)

k−2Y

j=0

[j+ 2|b|(1−γ)]

[1 +λ(k−1)]^(α_(β¹₁⁾₎^{k−1···(αq)k−1}

k−1···(βs)k−1(k+µ)(k+ 1 +µ), (k∈N^∗).

(3.1) Proof. Let f(z)∈ A be given by (1.1). Also let

g(z) = z+ X∞

k=2

b_kz^k ∈ SC^q,s_α,β(b, λ, γ), (3.2) so

a_k = (1 +µ)(2 +µ)

(k+µ)(k+ 1 +µ)b_k, (k ∈N^∗, µ∈R\(−∞,−1]) (3.3)

Thus, by using Theorem 1, we readily obtain

|a_k| ≤

(1 +µ)(2 +µ)

k−2Y

j=0

[j+ 2|b|(1−γ)]

[1 +λ(k−1)]^(α_(β¹₁⁾₎^{k−1···(αq)k−1}

k−1···(βs)k−1(k+µ)(k+ 1 +µ), (k ∈N^∗) which is precisely the assertion (3.1) of Theorem 2.

References

[1] O. Altintas, H.Irmak, S. Owa, H.M. Srivastava, Coefficient bounds for some families of starlike and convex functions of complex order,Applied Mathematics Letters 20(2007) 1218-1222

[2] J. Dziok, H. M. Srivastava, Classes of analytic functions associated with the generalized hypergeometric function, Appl. Math. Comput.

103(1999), no. 1, 1-13.

[3] J. Dziok, H.M. Srivastava, Certain subclasses of analytic functions as- sociated with the generalized hypergeometric function, Integral Trans- form. Spec. Funct. 14(2003) 7-18.

[4] M.A. Nasr, M.K. Aouf, Radius of convexity for the class of starlike func- tions of complex order, Bull. Fac. Sci. Assiut Univ. Sect. A 12(1983), 153-159.

[5] M.A. Nasr, M.K. Aouf, Bounded starlike functions of complex order, Proc. Indian Acad. Sci. Math. Sci. 92(1983) 97-102.

Sci. Math.25(1985) 1-12.

[7] P. Wiatrowski, On the coefficients of some family of holomorphic func- tions, Zeszyty Nauk. Uniw. odz Nauk. Mat.-Przyrod Ser. 39(2) (1970), 75-85.

Sevtap S¨umer Eker Dicle University

Department of Mathematics 21280 - Diyarbakır, Turkey

E-mail: [email protected] Bilal S¸eker Batman University

Department of Mathematics 72060 - Batman, Turkey

E-mail: [email protected] Mugur Alexandru Acu

”Lucian Blaga” University Department of Mathematics 550012 - Sibiu, Romania

E-mails: acu [email protected]