paper 17-18-2009. 97KB Jun 04 2011 12:03:07 AM

ACTA UNIVERSITATIS APULENSIS

No 18/2009

AN INTEGRAL OPERATOR ON ANALYTIC FUNCTIONS
WITH NEGATIVE COEFFICIENTS

Mugur Acu1 , Sevtap Sumer Eker
Abstract. In this paper we consider an integral operator on analytic
functions and prove some preserving theorems regarding some subclasses of
analytic functions with negative coefficients.
2000 Mathematics Subject Classification:30C45
Keywords and phrases: Function with negative coefficients, integral operators, S˘al˘agean operator
1. Introduction
Let H(U ) be the set of functions which are regular in the unit disc
U , A = {f ∈ H(U ) : f (0) = f ′ (0) − 1 = 0}, Hu (U ) = {f ∈ H(U ) :
f is univalent in U } and S = {f ∈ A : f is univalent in U }.
We denote with T the subset of the functions f ∈ S, which have the form
f (z) = z −

∞

X

aj z j , a j ≥ 0 , j ≥ 2 , z ∈ U

(1)

j=2

T ∗
T ∗
T c
∗
∗
c
and
with
T
=
T
S

,
T
(α)
=
T
S
(α),
T
=
T
S and T c (α) =
T c
T S (α), where 0 ≤ α < 1 .
Theorem 1.1 [5] For a function f having the form (1) the following
assertions are equivalents:
∞
X
(i)
jaj ≤ 1 ;
j=2

(ii) f ∈ T ;
1

Supported by GAR 19/2008

179

M. Acu, S.S. Eker - An integral operator on analytic functions with...
(iii) f ∈ T ∗ .
Regarding the classes T ∗ (α) and T c (α) with 0 ≤ α < 1 , we recall here
the following result:
Theorem 1.2 [5] A function f having the form (1) is in the class T ∗ (α)
if and only if:
∞
X
j−α
aj ≤ 1 ,
(2)
1−α

j=2
and is in the class T c (α) if and only if:
∞
X
j(j − α)
j=2

1−α

aj ≤ 1 .

(3)

Definition 1.1 [2] Let S ∗ (α, β) denote the class of functions having the
form (1) which are starlike and satisfy


zf ′ (z)



−1
f (z)


(4)
 zf ′ (z)