Paper-12-20-2009. 83KB Jun 04 2011 12:03:09 AM

ACTA UNIVERSITATIS APULENSIS

No 20/2009

A CONVEXITY PROPERTY FOR
AN INTEGRAL OPERATOR ON THE CLASS UST (K, γ)

Daniel Breaz and Shigeyoshi Owa
Abstract. An integral operator Fn (z) for analytic functions fi (z) in the open
unit disk U is introduced. The object of the present paper is to discuss a convexity
property for the integral operator Fn (z) involving k-uniformly starlike functions of
order γ.
2000 Mathematics Subject Classification: 30C45.
Keywords and Phrases: starlike, convex, uniformly starlike.
1. Introduction
Let A be the class of functions f (z) of the form
f (z) = z +

∞
X

aj z j

j=2

which are analytic in the open unit disc U = {z ∈ C : |z| < 1}.
Let S ∗ (α) denote the subclass of A consisting of functions f (z) which satisfy
 ′ 
zf (z)
Re
>α
(z ∈ U)
f (z)
for some α (0 ≦ α < 1). A function f (z) ∈ S ∗ (α) is said to be starlike of order α in
U. Also let K(α) be the subclass of A consisting of all functions f (z) which satisfy


zf ′′ (z)
Re 1 + ′
>α
(z ∈ U)

f (z)
for some α (0 ≦ α < 1). A function f (z) in K(α) is said to be convex of order α
in U. Further let UST (k, γ) be the subclass of A consisting of functions f (z) which
satisfy
 ′

 ′ 
 zf (z)

zf (z)
> k 
− 1  + γ
Re
(z ∈ U)
f (z)
f (z)
107

D. Breaz, S. Owa - A convexity property for an integral operator on the class...

for some k (k ≧ 0) and γ (0 ≦ γ < 1). A function f (z) ∈ UST (k, γ) is said to be
k-uniformly starlike of order γ. This class was introduced by Aghalary and Azadi
[1] (also see [3]).
For fi (z) ∈ A and αi > 0, we define the integral operator Fn (z) given by
 !
Z z Y
n 
fi (t) αi
Fn (z) =
dt.
t
0
i=1

This integral operator Fn (z) was first defined by Breaz and Breaz [2]. From the
definition for the integral operator, it is easy to see that Fn (z) ∈ A.

2. Main Result
Our main result for the integral operator Fn (z) is contained in
Theorem 1.

Let fi (z) ∈ UST (ki , γi ) for each i = 1, 2, 3, · · · , n. If αi > 0 (i =
1, 2, 3, · · · , n) and
n
X
0<
(1 − γi )αi ≦ 1,
i=1

then Fn (z) ∈ K(γ), where

γ =1−

n
X

(1 − γi )αi .

i=1

Proof. Since

Fn′ (z)

=


n 
Y
fi (z) αi
i=1

and
Fn′′ (z)

=

n
X
i=1

we obtain

αi



fi (z)
z

αi −1 

z

zfi′ (z) − fi (z)
zfi (z)
n

X
F ′′ (z)
=1+
αi

1 + n′
Fn (z)
i=1




 Y

n
fj (z) αj
,
z
j=1j6=i


zfi′ (z)
−1 .
fi (z)

Thus we see, for fi (z) ∈ U ST (ki , γi ),


 ′ 
n
n
X
X
Fn′′ (z)
zfi (z)
Re 1 + ′
=1 +
−
αi Re
αi
Fn (z)
fi (z)
i=1

108

i=1

D. Breaz, S. Owa - A convexity property for an integral operator on the class...

> 1+

n
X
i=1


  ′

n
X
 zfi (z)

αi ki 
− 1 + γi −

αi
fi (z)
i=1

≧ 1−

n
X

(1 − γi )αi .

i=1

This completes the proof of our theorem.
Corollary 2.

Let f (z) ∈ U ST (k, γ). If α > 0 and
0 < α ≦

then

F1 (z) =

Z

0

z



f (t)
t

1
,
1−γ
α

dt ∈ K(δ),

where δ = 1 − (1 − γ)α.
Remark 3. If we take α = 1 in the corollary, then the Alexander operator defined
by
Z z
f (t)
F (z) =
dt
t
0

is convex of order γ. Therefore, letting k = 0, we see that f (z) ∈ S ∗ (γ) implies that
Z z
f (t)
dt ∈ K(γ).
F (z) =
t
0
References

[1] R. Aghalary and Ch. Azadi, The Dziok-Srivastava operator and k-uniformly
starlike functions, J. Inequ. Pure Appl. Math. 6 (2005), 1 - 7.
[2] D. Breaz and N. Breaz, Two integral operators, Stu. Univ. Babes-Bolyai,
Mathematica, 3 (2002), 3 (2002), 13 - 21.
[3] A. W. Goodman, On uniformly starlike functions. J. Math. Anal. Appl. 155
(1991), 364 - 370
Daniel Breaz
Department of Mathematics
”1 Decembrie 1918” University
Alba Iulia, 510009, Alba
Romania
109

D. Breaz, S. Owa - A convexity property for an integral operator on the class...

e-mail: dbreaz@uab.ro

Shigeyoshi Owa
Department of Mathematics
Kinki University
Higashi-Osaka, Osaka 577-8502
Japan
e-mail : owa@math.kindai.ac.jp

110