paper 05-17-2009. 85KB Jun 04 2011 12:03:06 AM

ACTA UNIVERSITATIS APULENSIS

No 17/2009

THE INTEGRAL OPERATOR ON THE SP CLASS

Daniel Breaz, Nicoleta Breaz
Abstract. Let SP be the subclass of S consisting of all analytic and
univalent functions f (z) in the open unit disk U with f (0) = 0 and f ′ (0) = 1.
For fj (z) ∈ SP , an integral operator Fn (z) is introduced. The aim of the
present paper is to discuss the order of convexity for Fn (z).
2005 Mathematics Subject Classification: 30C45.
Kewwords and Phrases: analytic function, univalent function, integral operator, convex function.
1. Introduction
Let U = {z ∈ C, |z| < 1} be the unit disc of the complex plane and denote
by H (U ), the class of the holomorphic functions in U. Consider
n

A = f ∈ H (U ) , f (z) = z + a2 z 2 + a3 z 3 + ..., z ∈ U

o

be the class of analytic functions in U and S = {f ∈ A : f is univalent in U }.
Denote with K the class of the holomorphic functions in U with f (0) =
′
f (0) − 1 = 0, where is convex functions in U , defined by
zf ′′ (z)
K = f ∈ H (U ) : f (0) = f (0) − 1 = 0, Re
+ 1 > 0, z ∈ U .
f ′ (z)
(

(

′

)

)

A function f ∈ A is the convex function of order α, 0 ≤ α < 1 and denote

this class by K (α) if f verify the inequality
zf ′′ (z)
Re
+ 1 > α, z ∈ U .
f ′ (z)
)

(

43

D. Breaz, N. Breaz - The integral operator on the SP class
In the paper [2], F. Ronning introduced the class of univalent functions
denoted by SP . We say that the function f ∈ S is in SP if and only if





zf ′ (z)  zf ′ (z)

Re
>
− 1 ,
 f (z)

f (z)

(1)

for all z ∈ U .
The geometric interpretation of the relation (1) is that the class SP is the
class of all functions f ∈ S for which the expression zf ′ (z) /f (z) , z ∈ U takes
all values in the parabolic region
n

o

Ω = {ω : |ω − 1| ≤ Reω} = ω = u + iv : v 2 ≤ 2u − 1 .
We consider the integral operator defined in [1]
F (z) =

Z

f1 (t)
t

z

0

!α1

fn (t)
· ... ·
t

!αn

dt

(2)

and we study their properties.
Remark. We observe that for n = 1 and α1 = 1 we obtain the integral
operator of Alexander.
2. Main results
Theorem 1.Let αi , i ∈ {1, ..., n} be the real numbers with the properties
αi > 0 for i ∈ {1, ..., n},and
n
X

αi ≤ 1.

(3)

i=1

We suppose that the functions fi ∈ SP for i = {1, ..., n}. In this conditions
P
the integral operator defined in (2) is convex of order 1 − ni=1 αi .

Proof. We calculate for F the derivatives of the first and second order.
From (2) we obtain:
′

F (z) =

f1 (z)
z

!α1

44

fn (z)
· ... ·
z

!αn

D. Breaz, N. Breaz - The integral operator on the SP class

and
′′

F (z) =

n
X

fi (z)
z

αi

i=1

!αi −1

n
zfi′ (z) − fi (z) Y
fj (z)

zfi (z)
z
j=1

!

!αj

j6=i

zf1′ (z) − f1 (z)
zfn′ (z) − fn (z)
F ′′ (z)
=
α
+
...
+
α
.

1
n
F ′ (z)
zf1 (z)
zfn (z)
!

!

F ′′ (z)
fn′ (z) 1
f1′ (z) 1
+
...
+
α
.
=
α
−

−
n
1
F ′ (z)
f1 (z) z
fn (z) z
!

!

(4)

Multiply the relation (4) with z we obtain:
n
n
n
X
zfi′ (z) X
zfi′ (z)
zF ′′ (z) X

αi
αi
αi .
=
−1 =
−
F ′ (z)
fi (z)
fi (z)
i=1
i=1
i=1

!

(5)

The relation (5) is equivalent with
n
n
X
zfi′ (z) X
zF ′′ (z)
αi
αi + 1.
+1=
−
F ′ (z)
fi (z)
i=1
i=1

(6)

We calculate the real part from both terms of the above equality and obtain:
n
n
X
X
zF ′′ (z)
zfi′ (z)
Re
αi + 1.
−
αi Re
+1 =
F ′ (z)
fi (z)
i=1
i=1

!

!

(7)

Because fi ∈ SP for i = {1, ..., n} we apply in the above relation the
inequality (1) and obtain:
n
n
 zf ′ (z)
 X
X
zF ′′ (z)
 i

Re

α
+
1
>
−
1

αi + 1.
−
i
 fi (z)

F ′ (z)
i=1
i=1



!

 ′
 zf (z)






Because αi  fii(z) − 1 > 0 for all i ∈ {1, ..., n}, obtain that
n
X
zF ′′ (z)
αi .
+
1
>
1
−
Re
F ′ (z)
i=1

!

(8)

(9)

Using the hypothesis (3) in (9), we obtain that F is convex function of
P
order 1 − ni=1 αi .
45

D. Breaz, N. Breaz - The integral operator on the SP class
Remark. If

Pn

i=1

αi = 1 then
zF ′′ (z)
Re
+1 >0
F ′ (z)
!

(10)

so, F is the convex function.
Corollary 2. Let γ be the real numbers with the properties 0 < γ < 1.
We suppose that the functions f ∈ SP . In this conditions the integral operator
γ
R
F (z) = 0z f (z)
dt is convex of order 1 − γ.
z
Proof. In the Theorem 1, we consider n = 1, α1 = γ and f1 = f .

Theorem 3. We suppose that the function f ∈ SP . In this condition the
integral operator of Alexander defined by
FA (z) =

Z

0

z

f (t)
dt
t

(11)

is convex.
Proof. We have:
FA′ (z) =
and

f (z) ′′
zf ′ (z) − f (z)
, FA (z) =
z
zf (z)
zf ′ (z)
zFA′′ (z)
=
− 1.
FA′ (z)
f (z)

(12)

From (12) we have:





zFA′′ (z)
zf ′ (z)  zf ′ (z)

Re
+
1
=
Re
>
−
1

 > 0.
 f (z)

FA′ (z)
f (z)
!

(13)

So, the relation (13) imply that the Alexander operator FA is convex.
Remark. Theorem 3 can be obtained from the Corollary 2 for γ = 1.

46

D. Breaz, N. Breaz - The integral operator on the SP class
References
[1] D. Breaz and N. Breaz, Two integral operators, Studia Universitatis
Babeş-Bolyai, Mathematica, Cluj Napoca, No.3(2002), 13-21
[2] F. Ronning, Uniformly convex functions and a corresponding class of
starlike functions, Proc. Amer. Math. Soc., 118(1993), 190-196.
Authors:
Daniel Breaz
Department of Mathematics
” 1 Decembrie 1918 ” University
Alba Iulia, Romania
e-mail: dbreaz@uab.ro
Nicoleta Breaz
Department of Mathematics
” 1 Decembrie 1918 ” University
Alba Iulia, Romania
e-mail: nbreaz@uab.ro

47