A Class of Harmonic Multivalent Functions Defined by an Integral Operator

www.i-csrs.org

Available free online at http://www.geman.in

A Class of Harmonic Multivalent Functions Defined by an Integral Operator

R. Ezhilarasi¹, T.V. Sudharsan² and K.G. Subramanian³

1,2Department of Mathematics, SIVET College Chennai - 600 073, India

1E-mail: [email protected]

2E-mail: [email protected]

3School of Computer Sciences Universiti Sains Malaysia

11800 Penang, Malaysia E-mail: [email protected] (Received: 27-1-14 / Accepted: 12-3-14)

Abstract

A new class of harmonic multivalent functions defined by an integral oper- ator is introduced. Coefficient inequalities, extreme points, distortion bounds, inclusion results and closure under an integral operator for this class are ob- tained.

Keywords: Harmonic functions, multivalent functions, integral operator.

1 Introduction

Harmonic mappings are important in different applied fields of study [1]. Har- monic mappings in a simply connected domain D⊆C are univalent complex valued harmonic functions f =u+iv where both u and v are real harmonic inD.

Let S_H denote the family of harmonic functions f = h+g [6], which are univalent and sense-preserving in the open unit disc ∆ ={z : |z| <1} where hand g are analytic inDand f is normalized byf(0) =h(0) =f_z(0)−1 = 0.

Subclasses of harmonic functions have been studied by many authors (See for

example, Aouf et al. [2], Atshan and Kulkarni [3], Chandrashekar et al. [5], Cotˆırl˘a [7], Jahangiri [9, 10], Jahangiri and Ahuja [11], Jahangiri et al. [12]).

The classHp(n) (p, n ∈N ={1,2, . . .}), consisting of allp-valent harmonic functionsf =h+g that are orientation preserving in ∆ was defined by Ahuja and Jahangiri [11] where h and g are of the form

h(z) = z^p+

∞

X

k=2

ak+p−1z^k+p−1, g(z) =

∞

X

k=1

bk+p−1z^k+p−1, |b_p|<1. (1) An integral operator Iⁿ was introduced by Salagean [14] which is given below in a slightly modified form as stated by [7].

(i) I⁰f(z) =f(z);

(ii) I¹f(z) =If(z) =pRz

0 f(t)t⁻¹dt;

(iii) Iⁿf(z) = I(Iⁿ⁻¹f(z)),n ∈N,f ∈A

where A={f ∈H :f(z) = z+a₂z²+. . .}and H =H(∆), the class of holomorphic functions in ∆.

The modified Salagean integral operator off =h+g given by (1) is defined [7] as

Iⁿf(z) = Iⁿh(z) + (−1)ⁿIⁿg(z), (2) where

Iⁿh(z) =z^p+

∞

X

k=2

p k+p−1

n

ak+p−1z^k+p−1 and Iⁿg(z) =

∞

X

k=1

p k+p−1

n

bk+p−1z^k+p−1

For 0≤β < 1, 0≤t ≤1, n ∈ N, z ∈∆, let H_p(n, β, t) denote the family of harmonic functions of the form (1) such that

Re

Iⁿf(z)

(1−t)z^p+tIⁿ⁺¹f(z)

> β (3)

whereIⁿ is defined by (2).

Let H_p(n, β, t) denote the subclass consists of harmonic functions f_n = h+g_n inH_p(n, β, t) so that h and g_n are of the form

h(z) =z^p−

∞

X

k=2

ak+p−1z^k+p−1 and g_n(z) = (−1)ⁿ⁻¹

∞

X

k=1

bk+p−1z^k+p−1 (4) whereak+p−1, bk+p−1 ≥0 and |b_p|<1.

Remark 1.1 The class H_p(n, β, t) reduces to the class H_p(n, β) [7] and to the class H_p(n+ 1, n, β,0) [8], when t= 1.

Coefficient inequalities, extreme points, distortion bounds, inclusion results and closure under an integral operator for functions in the classH_p(n, β, t) are obtained.

2 Main Results

A sufficient coefficient condition for harmonic functions belonging to the class Hp(n, β, t) is now derived.

Theorem 2.1 Let f =h+g be given by (1). If

∞

X

k=2

φ(n, p, k, β, t)|ak+p−1|+

∞

X

k=1

ψ(n, p, k, β, t)|bk+p−1| ≤1 (5)

where

φ(n, p, k, β, t) = p

k+p−1

nh 1−βt

p k+p−1

i

1−β ψ(n, p, k, β, t) =

p k+p−1

nh

1 +βt

p k+p−1

i

1−β ,

0≤β <1, 0≤t≤1, n∈N. Then f ∈H_p(n, β, t).

Proof. To show that f ∈ H_p(n, β, t) according to the condition (3), we only need to show that if (5) holds, then

Re

Iⁿf(z)

(1−t)z^p+tIⁿ⁺¹f(z)

=ReA(z) B(z) ≥β wherez =re^iθ, 0≤θ ≤2π, 0≤r <1 and 0≤β <1.

Note thatA(z) =Iⁿf(z) and

B(z) = (1−t)z^p+tIⁿ⁺¹f(z).

Using the fact thatRe w≥β if and only if|1−β+w| ≥ |1 +β−w|, it suffices to show that

|A(z) + (1−β)B(z)| − |A(z)−(1 +β)B(z)| ≥0 (6)

SubstitutingA(z) and B(z) in (6) we obtain

|A(z) + (1−β)B(z)| − |A(z)−(1 +β)B(z)|

=

Iⁿf(z) + (1−β)[(1−t)z^p+tIⁿ⁺¹f(z)]

−

Iⁿf(z)−(1 +β)[(1−t)z^p+tIⁿ⁺¹f(z)]

=

z^p+

∞

X

k=2

p k+p−1

n

ak+p−1z^k+p−1 + (−1)ⁿ

∞

X

k=1

p k+p−1

n

bk+p−1z^k+p−1

+ (1−β)

"

(1−t)z^p+tz^p+t

∞

X

k=2

p k+p−1

n+1

ak+p−1z^k+p−1

+t(−1)ⁿ⁺¹

∞

X

k=1

p k+p−1

n+1

bk+p−1z^k+p−1

#

−

z^p+

∞

X

k=2

p k+p−1

n

ak+p−1z^k+p−1+ (−1)ⁿ

∞

X

k=1

p k+p−1

n

bk+p−1z^k+p−1

−(1 +β)

"

(1−t)z^p+tz^p+t

∞

X

k=2

p k+p−1

n+1

ak+p−1z^k+p−1

+t(−1)ⁿ⁺¹

∞

X

k=1

p k+p−1

n+1

bk+p−1z^k+p−1

#

=

(2−β)z^p+

∞

X

k=2

p k+p−1

n

1 + (1−β)t

p k+p−1

a_k+p−1z^k+p−1

−(−1)ⁿ⁺¹

∞

X

k=1

p k+p−1

n

1−(1−β)t

p k+p−1

bk+p−1z^k+p−1

−

−βz^p+

∞

X

k=2

p k+p−1

n

1−(1 +β)t

p k+p−1

ak+p−1z^k+p−1

−(−1)ⁿ⁺¹

∞

X

k=1

p k+p−1

n

1 + (1 +β)t

p k+p−1

b_k+p−1z^k+p−1

≥(2−β)|z|^p−

∞

X

k=2

p k+p−1

n

1 + (1−β)t

p k+p−1

|ak+p−1||z|^k+p−1

−

∞

X

k=1

p k+p−1

n

1−(1−β)t

p k+p−1

|bk+p−1||z|^k+p−1

−β|z|^p −

∞

X

k=2

p k+p−1

n

1−(1 +β)t

p k+p−1

|ak+p−1||z|^k+p−1

−

∞

X

k=1

p k+p−1

n

1 + (1 +β)t

p k+p−1

|bk+p−1||z|^k+p−1

≥2(1−β)|z|^p −

∞

X

k=2

p k+p−1

n

1 + (1−β)t

p k+p−1

+1−(1 +β)t

p k+p−1

|ak+p−1||z|^k+p−1

−

∞

X

k=1

p k+p−1

n

1−(1−β)t

p k+p−1

+1 + (1 +β)t

p k+p−1

|bk+p−1||z|^k+p−1

≥2(1−β)|z|^p −

∞

X

k=2

2

p k+p−1

n 1−βt

p k+p−1

|ak+p−1||z|^k+p−1

−

∞

X

k=1

2

p k+p−1

n 1 +βt

p k+p−1

|bk+p−1||z|^k+p−1

= 2(1−β)|z|^p



1−







∞

X

k=2

p k+p−1

nh

1−βt

p k+p−1

i

1−β |ak+p−1||z|^k−1

+

∞

X

k=1

p k+p−1

nh

1 +βt

p k+p−1

i

1−β |bk+p−1||z|^k−1











≥2(1−β)



1−







∞

X

k=2

p k+p−1

nh

1−βt

p k+p−1

i

1−β |a_k+p−1|

+

∞

X

k=1

p k+p−1

nh

1 +βt

p k+p−1

i

1−β |bk+p−1|











≥0, by (5).

This completes the proof.

The harmonic univalent functions f(z) =z^p +

∞

X

k=2

1

φ(n, p, k, β, t)x_kz^k+p−1+

∞

X

k=1

1

ψ(n, p, k, β, t)y_kz^k+p−1 (7) wheren ∈N and

∞

X

k=2

|x_k|+

∞

X

k=1

|y_k|= 1, shows that the coefficient bound given by (5) is sharp.

This is because

∞

X

k=2

φ(n, p, k, β, t)|a_k+p−1|+

∞

X

k=1

ψ(n, p, k, β, t)|b_k+p−1|

=

∞

X

k=2

φ(n, p, k, β, t) 1

φ(n, p, k, β, t)|X_k|+

∞

X

k=1

ψ(n, p, k, β, t) 1

ψ(n, p, k, β, t)|Y_k|

=

∞

X

k=2

|X_k|+

∞

X

k=1

|Y_k|= 1.

We now show that the condition (5) is also necessary for functions fn = h+g_n, where h and g_n are of the form (4).

Theorem 2.2 Let fn = h+gn be given by (4). Then fn ∈ Hp(n, β, t) if and only if

∞

X

k=2

φ(n, p, k, β, t)ak+p−1+

∞

X

k=1

ψ(n, p, k, β, t)bk+p−1 ≤1. (8)

where 0≤β <1, 0≤t ≤1, n ∈N, with b_k+p−1 > a_k+p−1, for every k ≥2.

Proof. We only need to prove the “only if” part of the theorem because H_p(n, β, t) ⊂ H_p(n, β, t). To this end, for functions f_n of the form (4), we notice that the condition

Re

Iⁿf(z)

(1−t)z^p+tIⁿ⁺¹f(z)

> β is equivalent to

Re

































 h

(1−β)z^p−

∞

X

k=2

p k+p−1

n 1−βt

p k+p−1

ak+p−1z^k+p−1 +(−1)²ⁿ⁻¹

∞

X

k=1

p k+p−1

n 1 +βt

p k+p−1

bk+p−1z^k+p−1i h

z^p −t

∞

X

k=2

p k+p−1

n+1

ak+p−1z^k+p−1 +t(−1)²ⁿ

∞

X

k=1

p k+p−1

n+1

bk+p−1z^k+p−1 i



































≥0 (9)

We observe that the above required condition (9) must hold for all values ofz in ∆. Choosing the values ofz on the positive real axis where 0≤z =r <1, we have for bk+p−1 > ak+p−1, for every k ≥2,

h

(1−β)−

∞

X

k=2

p k+p−1

n 1−βt

p k+p−1

ak+p−1r^k−1

−

∞

X

k=1

p k+p−1

n 1 +βt

p k+p−1

bk+p−1r^k−1i h

1−t

∞

X

k=2

p k+p−1

n+1

ak+p−1r^k−1 +t

∞

X

k=1

p k+p−1

n+1

bk+p−1r^k−1i

≥0 (10)

If the condition (8) does not hold, then the expression in (10) is negative for r sufficiently close to 1. Hence there exist z0 = r0 in (0, 1) for which the quotient in (10) is negative. This contradicts the required condition for f_n ∈H_p(n, β, t) and this completes the proof.

The extreme points of closed convex hull ofH_p(n, β, t), denoted byclco H_p(n, β, t) is now determined.

Theorem 2.3 Let f_n be given by (4). Then f_n ∈H_p(n, β, t) if and only if

f_n(z) =

∞

X

k=1

[xk+p−1hk+p−1(z) +yk+p−1g_nk+p−1(z)],

where h_p(z) = z^p, hk+p−1(z) = z^p− 1

φ(n, p, k, β, t)z^k+p−1, k = 2,3, . . ., and g_nk+p−1(z) = z^p+ (−1)ⁿ⁻¹ 1

ψ(n, p, k, β, t)z^k+p−1, k = 1,2,3, . . .. x_k+p−1 ≥0, y_k+p−1 ≥0, x_p = 1−

∞

X

k=2

x_k+p−1−

∞

X

k=1

y_k+p−1.

In particular, the extreme point ofH_p(n, β, t) are{hk+p−1} and {g_nk+p−1}.

Proof. Suppose

fn(z) =

∞

X

k=1

[xk+p−1hk+p−1(z) +yk+p−1gnk+p−1(z)]

=

∞

X

k=1

(xk+p−1+yk+p−1)z^p−

∞

X

k=2

1

φ(n, p, k, β, t)xk+p−1z^k+p−1 + (−1)ⁿ⁻¹

∞

X

k=1

1

ψ(n, p, k, β, t)yk+p−1z^k+p−1

=z^p−

∞

X

k=2

1

φ(n, p, k, β, t)x_k+p−1z^k+p−1 + (−1)ⁿ⁻¹

∞

X

k=1

1

ψ(n, p, k, β, t)yk+p−1z^k+p−1

Then

∞

X

k=2

φ(n, p, k, β, t)|a_k+p−1|+

∞

X

k=1

ψ(n, p, k, β, t)|b_k+p−1|

=

∞

X

k=2

φ(n, p, k, β, t)

1

φ(n, p, k, β, t)xk+p−1

+

∞

X

k=1

ψ(n, p, k, β, t)

1

ψ(n, p, k, β, t)yk+p−1

=

∞

X

k=2

xk+p−1 +

∞

X

k=1

yk+p−1 = 1−x_p ≤1.

and sofn(z)∈clco Hp(n, β, t).

Conversely, iff_n(z)∈clco H_p(n, β, t). Letting

x_p = 1−

∞

X

k=2

xk+p−1−

∞

X

k=1

yk+p−1.

Set

xk+p−1 =φ(n, p, k, β, t)ak+p−1, k= 2,3, . . . and yk+p−1 =ψ(n, p, k, β, t)bk+p−1, k= 1,2, . . . .

The required representations is obtained as f_n(z) = z^p−

∞

X

k=2

ak+p−1z^k+p−1+ (−1)ⁿ⁻¹

∞

X

k=1

bk+p−1z^k+p−1

=z^p−

∞

X

k=2

1

φ(n, p, k, β, t)xk+p−1z^k+p−1 + (−1)ⁿ⁻¹

∞

X

k=1

1

ψ(n, p, k, β, t)yk+p−1z^k+p−1

=z^p−

∞

X

k=2

[z^p −hk+p−1(z)]xk+p−1−

∞

X

k=1

[z^p−gk+p−1(z)]yk+p−1

=

"

1−

∞

X

k=2

x_k+p−1−

∞

X

k=1

y_k+p−1

# z^p

+

∞

X

k=2

xk+p−1hk+p−1(z) +

∞

X

k=1

yk+p−1g_nk+p−1(z)

=

∞

X

k=1

[xk+p−1hk+p−1(z) +yk+p−1g_nk+p−1(z)]

We now obtain the distortion bounds for functions in H_p(n, β, t).

Theorem 2.4 Let f_n∈H_p(n, β, t). Then for |z|=r <1 we have

|f_n(z)| ≤(1 +b_p)r^p+{θ(n, p, k, β, t)−Ω(n, p, k, β, t)b_p}r^n+p+1 and

|fn(z)| ≥(1−bp)r^p− {θ(n, p, k, β, t)−Ω(n, p, k, β, t)bp}r^n+p+1 where

θ(n, p, k, β, t) = 1−βt p

p+1

nh 1−

p p+1

βti Ω(n, p, k, β, t) = 1 +βt

p p+1

nh 1−

p p+1

βti

We prove the right hand side inequality for|f_n|. The proof for the left hand inequality is similar. Let f_n ∈H_p(n, β, t) taking the absolute value of f_n then

by Theorem 2.2, we obtain:

|f_n(z)|=

z^p−

∞

X

k=2

a_k+p−1z^k+p−1+ (−1)ⁿ⁻¹

∞

X

k=1

b_k+p−1z^k+p−1

≤r^p+

∞

X

k=2

ak+p−1r^k+p−1+

∞

X

k=1

bk+p−1r^k+p−1

=r^p +b_pr^p+

∞

X

k=2

(ak+p−1+bk+p−1)r^k+p−1

≤r^p+b_pr^p+

∞

X

k=2

(ak+p−1+bk+p−1)r^p+1

= (1 +bp)r^p+θ(n, p, k, β, t)

∞

X

k=2

1

θ(n, p, k, β, t)(ak+p−1+bk+p−1)r^p+1

≤(1 +bp)r^p +θ(n, p, k, β, t)r^n+p+1

" _∞ X

k=2

φ(n, p, k, β, t)ak+p−1+ψ(n, p, k, β, t)bk+p−1

#

≤(1 +bp)r^p +{θ(n, p, k, β, t)−Ω(n, p, k, β, t)bp}r^n+p+1.

3 Closure Property of the Class H

(n, β, t)

In the next two theorems, we prove that the classH_p(n, β, t) is invariant under convolution and convex combinations of its members.

The convolution of two harmonic functions, fn(z) =z^p −

∞

X

k=2

ak+p−1z^k+p−1+ (−1)ⁿ⁻¹

∞

X

k=1

bk+p−1z^k+p−1 (11) and

F_n(z) = z^p−

∞

X

k=2

Ak+p−1z^k+p−1+ (−1)ⁿ⁻¹

∞

X

k=1

Bk+p−1z^k+p−1 (12) is defined as

(f_n∗F_n)(z) =f_n(z)∗F_n(z)

=z^p−

∞

X

k=2

ak+p−1Ak+p−1z^k+p−1+ (−1)ⁿ⁻¹

∞

X

k=1

bk+p−1Bk+p−1z^k+p−1 (13) Using this definition, we first show that the class H_p(n, β, t) is closed under convolution.

Theorem 3.1 For 0 ≤ α ≤ β < 1, 0 ≤ t ≤ 1, let f_n ∈ H_p(n, β, t) and F_n∈H_p(n, α, t). Then

f_n∗F_n ∈H_p(n, β, t)⊂H_p(n, α, t).

Proof. Let f_n(z) = z^p −

∞

X

k=2

ak+p−1z^k+p−1 + (−1)ⁿ⁻¹

∞

X

k=1

bk+p−1z^k+p−1 be in Hp(n, β, t) and Fn(z) =z^p−

∞

X

k=2

Ak+p−1z^k+p−1+ (−1)ⁿ⁻¹

∞

X

k=1

Bk+p−1z^k+p−1 be inH_p(n, α, t).

Then the convolution fn∗Fn is given by (13). We wish to show that the coefficients off_n∗F_nsatisfy the required condition given in Theorem 2.2. For F_n ∈ H_p(n, α, t), we note that Ak+p−1 < 1 and Bk+p−1 < 1. Now, for the convolution functionfn∗Fn, we obtain

∞

X

k=2

φ(n, p, k, α, t)ak+p−1Ak+p−1+

∞

X

k=1

ψ(n, p, k, α, t)bk+p−1Bk+p−1

≤

∞

X

k=2

φ(n, p, k, α, t)a_k+p−1+

∞

X

k=1

ψ(n, p, k, α, t)b_k+p−1

≤

∞

X

k=2

φ(n, p, k, β, t)ak+p−1+

∞

X

k=1

ψ(n, p, k, β, t)bk+p−1

≤1,

since 0≤α≤β <1 and f_n ∈H_p(n, β, t).

Now, we show that Hp(n, β, t) is closed under convex combination of its members.

Theorem 3.2 The class H_p(n, β, t) is closed under convex combination.

Proof. Fori= 1,2,3, . . .. Suppose f_ni ∈H_p(n, β, t), where f_ni is given by f_ni(z) =z^p −

∞

X

k=2

ai,k+p−1z^k+p−1+ (−1)ⁿ⁻¹

∞

X

k=1

bi,k+p−1z^k+p−1.

Then by (8)

∞

X

k=2

φ(n, p, k, β, t)ai,k+p−1+

∞

X

k=1

ψ(n, p, k, β, t)bi,k+p−1 ≤1 (14)

For

∞

X

i=1

ti = 1, 0≤ti ≤1, the convex combination of fni may be written as

∞

X

i=1

t_if_ni(z) =z^p−

∞

X

k=2

∞

X

i=1

t_ia_i,k+p−1

!

z^k+p−1

+ (−1)ⁿ⁻¹

∞

X

k=1

∞

X

i=1

t_ibi,k+p−1

!

z^k+p−1 Using the inequality (14), we obtain

∞

X

k=2

φ(n, p, k, β, t)

∞

X

i=1

t_iai,k+p−1

! +

∞

X

k=1

ψ(n, p, k, β, t)

∞

X

i=1

t_ibi,k+p−1

!

=

∞

X

i=1

t_i

∞

X

k=2

φ(n, p, k, β, t)ai,k+p−1+

∞

X

k=1

ψ(n, p, k, β, t)bi,k+p−1

!

≤

∞

X

i=1

t_i = 1,

which is the required coefficient condition.

Finally, we examine the closure property of the class H_p(n, β, t) under the generalized Bernardi-Libera-Livingston integral operator (see [4, 13]) L_c(f) which is defined by,

Lc(f) = c+p z^c

Z z 0

t^c−1f(t)dt, c >−1.

Theorem 3.3 Let f_n(z)∈H_p(n, β, t). Then L_c(f(z))∈H_p(n, β, t).

Proof. From the representation of L_c(f_n(z)), it follows that L_c(f_n(z)) = c+p

z^c Z z

0

t^c−1

"

t^p −

∞

X

k=2

a_k+p−1t^k+p−1+ (−1)ⁿ⁻¹

∞

X

k=1

b_k+p−1t^k+p−1

# dt

=z^p−

∞

X

k=2

c+p

c+p+k−1ak+p−1z^k+p−1 + (−1)ⁿ⁻¹

∞

X

k=1

c+p

c+p+k−1bk+p−1z^k+p−1

=z^p−

∞

X

k=2

Xk+p−1z^k+p−1+ (−1)ⁿ⁻¹

∞

X

k=1

Yk+p−1z^k+p−1

where

Xk+p−1 = c+p

c+p+k−1ak+p−1 and Yk+p−1 = c+p

c+p+k−1bk+p−1. Hence

∞

X

k=2

φ(n, p, k, β, t) c+p

c+p+k−1ak+p−1+

∞

X

k=1

ψ(n, p, k, β, t) c+p

c+p+k−1bk+p−1

≤

∞

X

k=2

φ(n, p, k, β, t)ak+p−1+

∞

X

k=1

ψ(n, p, k, β, t)bk+p−1

≤1 by (8).

Hence by Theorem 2.2,L_c(f_n(z))∈H_p(n, β, t).

Acknowledgements: The authors are grateful to the reviewer for useful comments.

References

[1] O.P. Ahuja, Planar harmonic univalent and related mappings, JIPAM, 6(4)(Article 122) (2005), 18.

[2] M.K. Aouf, A.O. Mostafa, A.A. Shamandy and A.K. Wagdy, A study on certain class of harmonic functions of complex order associated with convolution, Le Matematiche, LXVI(Fist. 11) (2012), 169-183.

[3] W.G. Atshan and S.R. Kulkarni, New classes of multivalently harmonic functions,Int. Journal of Math. Analysis, 2(3) (2008), 111-121.

[4] S.D. Bernardi, Convex and starlike univalent function, Trans. Amer.

Math. Soc., 135(1969), 429-446.

[5] R. Chandrashekar, G. Murugusundaramoorthy, S. K. Lee and K.G. Sub- ramanian, A class of complex-valued harmonic functions defined by Dziok- Srivastava operator,Chamchuri Journal of Mathematics, 1(2) (2009), 31- 42.

[6] J. Clunie and T. Sheil-Small, Harmonic univalent functions, Ann. Acad.

Sci. Fenn., Ser. A.I., Math., 9(1984), 3-25.

[7] L.I. Cotˆırl˘a, Harmonic multivalent functions defined by integral operator, Studia Univ. Babe¸s - Bolyai Mathematica, LIV(1) (2009), 65-74.

[8] L.I. Cotˆırl˘a, A new class of Harmonic multivalent functions defined by an integral operator,Acta Universitatis Apulensis, 21(2010), 55-63.

[9] J.M. Jahangiri, Coefficient bounds and univalence criteria for harmonic functions with negative coefficients,Ann. Uni. Mariae Curie-Sklodowska, 52(1998), 57-66.

[10] J.M. Jahangiri, Harmonic functions starlike in the unit disc, J. Math.

Anal. Appl., 235(1999), 470-477.

[11] J.M. Jahangiri and O.P. Ahuja, Multivalent harmonic starlike functions, Ann. Univ. Marie Curie-Sklodowska (Sect. A), LVI(2001), 1-13.

[12] J.M. Jahangiri, Y.C. Kim and H.M. Srivastava, Construction of a certain class of harmonic close-to-convex function associated with the Alexander integral transform, Int. Trans. and Special Func., 14(3) (2003), 237-242.

[13] R.J. Libera, Some classes of regular univalent functions, Proc. Amer.

Math. Soc., 16(1965), 755-758.

[14] G.S. Salagean, Subclass of univalent functions, Lecture Notes in Math., Springer-Verlag, 1013(1983), 362-372.