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ACTA UNIVERSITATIS APULENSIS

No 16/2008

THE INTEGRAL OPERATOR ON THE SP (α, β) CLASS

Daniel Breaz

1

Abstract. In this paper we present a convexity condition for a integral
operator F defined in formula (2) on the class SP (α, β).
2000 Mathematics Subject Classification: 30C45
Keywords and phrases: Univalent functions, integral operator, convex
functions.
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 olomorphic functions in U. Consider A =
{f ∈ H (U ) , f (z) = z + a2 z 2 + a3 z 3 + ..., z ∈ U } be the class of analytic functions in U and S = {f ∈ A : f is univalent in U }.
Denote with K the class of the olomorphic functions in U with f (0) =
f ′ (0) − 1 = 0, where is convex functions in U , defined by




 ′′
zf (z)
′
+ 1 > 0, z ∈ U
K = f ∈ H (U ) : f (0) = f (0) − 1 = 0, Re
f ′ (z)
and denote for K (µ) the functions convex by the order µ, µ ∈ [0, 1), defined
by

K (µ) =

1



′

f ∈ H (U ) : f (0) = f (0) − 1 = 0, Re

Supported by GAR 19/2008

17





zf ′′ (z)
+ 1 > µ, z ∈ U .
f ′ (z)

D. Breaz - The integral operator on the SP (α, β) class
In the paper (3) F. Ronning introduced the class of univalent functions
SP (α, β) , α > 0, β ∈ [0, 1), the class of all functions f ∈ S which have the
property:
 ′

′

 zf (z)


 ≤ Re zf (z) + α − β,
−
(α
+
β)
(1)
 f (z)

f (z)
for all z ∈ U .
Geometric interpretation: f ∈ SP (α, β) if and only if zf ′ (z) /f (z) , z ∈
U takes all values in the parabolic region



Ωα,β = {ω : |ω − (α + β)| ≤ Reω + α − β} = ω = u + iv : v 2 ≤ 4α (u − β) .
We consider the integral operator defined in [2]


α
α
Z z
fn (t) n
f1 (t) 1
· ... ·
F (z) =
dt
t
t
0

(2)

and we study your 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} the real numbers with the properties
αi > 0 for i ∈ {1, ..., n},
n
X
i=1

Pn

αi <

1
α−β+1

(3)

and (β − α − 1) i=1 αi + 1 ∈ (0, 1). We suppose that the functions fi ∈
SP (α, β) for i = {1, ..., n} and α > 0, β ∈ [0, 1). In this conditions
the
P

integral operator defined in (2) is convex by the order (β − α − 1) ni=1 αi +1.
Proof. We calculate for F the derivatives of the first and second order.
From (2) we obtain:
18

D. Breaz - The integral operator on the SP (α, β) class

′

F (z) =
and

′′

F (z) =

n
X

αi

i=1

F ′′ (z)
= α1
F ′ (z)







fi (z)
z

f1 (z)
z

α 1

αi −1 



f1′ (z) 1
−
f1 (z) z





n



αn

Y

n 
j=1

fj (z)
z

j6=i

α j

+ ... + αn



zfn′ (z) − fn (z)
zfn (z)

+ ... + αn



fn′ (z) 1
−
fn (z) z

Multiply the relation (4) with z we obtain:
zF ′′ (z) X
=
αi
F ′ (z)
i=1

fn (z)
z

zfi′ (z) − fi (z)
zfi (z)

zf1′ (z) − f1 (z)
zf1 (z)

F ′′ (z)
= α1
F ′ (z)

· ... ·







.

.

(4)

 X
n
n
zfi′ (z)
zf ′ (z) X
−1 =
−
αi i
αi .
fi (z)
f
(z)
i
i=1
i=1

(5)

The relation (5) is equivalent with
n

zF ′′ (z) X
=
αi
F ′ (z)
i=1

and

n

X
zF ′′ (z)
+
1
=
αi
F ′ (z)
i=1





zfi′ (z)
+α−β
fi (z)

zfi′ (z)
+α−β
fi (z)





+ (β − α − 1)

+ (β − α − 1)

n
X

αi .

(6)

αi + 1.

(7)

i=1

n
X
i=1

We calculate the real part from both terms of the above equality and
obtain:

Re



( n

)
n
X
X  zf ′ (z)
zF ′′ (z)
i
+(β − α − 1)
+ 1 = Re
+α−β
αi +1.
αi
F ′ (z)
f
(z)
i
i=1
i=1
(8)
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D. Breaz - The integral operator on the SP (α, β) class
Because fi ∈ SP (α, β) for i = {1, ..., n} we apply in the above relation
the inequality (1) and obtain:

Re




 ′
 X
n
n
X

 zfi (z)
zF ′′ (z)


+1 ≥
− (α + β) +(β − α − 1)
αi +1. (9)
αi 
F ′ (z)
f
(z)
i
i=1
i=1


 ′
 zf (z)

Because αi  fii(z) − (α + β) > 0 for all i ∈ {1, ..., n} and the inequality
(3) we obtain that


n
X
zF ′′ (z)
Re
+ 1 ≥ (β − α − 1)
αi + 1 > 0.
(10)
F ′ (z)
i=1
P
From (10) and because (β − α − 1) ni=1 αi +1 ∈ (0, 1) we obatin
Pnthat the
integral operator defined in (2) is convex by the order (β − α − 1) i=1 αi +1.


1
.
Corollary 2. Let γ the real numbers with the properties 0 < γ < α−β+1
We suppose that the functions f ∈ SP (α, 
β) and α > 0, β ∈ [0, 1). In this
R z f (z) γ
conditions the integral operator F (z) = 0
dt is convex.
z

Proof. In the Theorem 1, we consider n = 1, α1 = γ and f1 = f .

For α = β ∈ (0, 1) we obtain the class S (α, α) where is caracterized by
the next property

 ′

 zf (z)
zf ′ (z)


(11)
 f (z) − 2α ≤ Re f (z) .

Corollary 3. Let αi , i ∈ {1, ..., n} the real numbers with the properties
αi > 0 for i ∈ {1, ..., n} and
1−

n
X

αi ∈ [0, 1) .

(12)

i=1

We suppose that the functions fi ∈ SP (α, α) for i = {1, ..., n} and α ∈
(0, 1). In this
Pn conditions the integral operator defined in (2) is convex by the
order 1 − i=1 αi .
Proof. From (2) obtain that

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D. Breaz - The integral operator on the SP (α, β) class

n

n

zF ′′ (z) X zfi′ (z) X
=
−
αi
αi ,
F ′ (z)
fi (z)
i=1
i=1

(13)

where is equivalent with
Re



 X
n
n
zfi′ (z) X
zF ′′ (z)
+
1
=
−
α
Re
αi + 1,
i
F ′ (z)
f
(z)
i
i=1
i=1

(14)

From (11) and (14) obtain that:


 ′
 X
n
n
X

 zfi (z)
zF ′′ (z)


+1 >
− 2α + 1 −
Re
αi 
αi .
(15)
F ′ (z)
fi (z)
i=1
i=1
 ′

Pn
 zfi (z)

Because
α
−
2α

 > 0 for all i ∈ {1, ..., n}, from (15) we
i=1 i fi (z)
obatin:


Re




n
X
zF ′′ (z)
+1 >1−
αi .
F ′ (z)
i=1

(16)

Now, fromP
(12) we obtain that the operator defined in (2) is convex by
the order 1 − ni=1 αi .


For α = β = 12 we observe that SP 12 , 21 = SP . This class is defined by
Ronning in the paper [4]. For the class SP we have the next result:
Corollary 4.Let αi , i ∈ {1, ..., n} the real numbers with the properties
αi > 0 for i ∈ {1, ..., n} and
1−

n
X

αi ∈ [0, 1) .

(17)

i=1

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

21

D. Breaz - The integral operator on the SP (α, β) class
References
[1] M. Acu, Operatorul integral Libera-Pascu si proprietatile acestuia cu
privire la functiile uniform stelate, convexe, aproape convexe si α-uniform
convexe,Editura Universitatii ”Lucian Blaga” din Sibiu, 2005.
[2] D. Breaz, N. Breaz, Two integral operators, Studia Universitatis Babe¸s
-Bolyai, Mathematica, Cluj Napoca, No. 3-2002,pp. 13-21.
[3] F. Ronning, Integral reprezentations of bounded starlike functions,
Ann. Polon. Math., LX, 3(1995), 289-297.
[4] F. Ronning, Uniformly convex functions and a corresponding class of
starlike functions, Proc. Amer. Math. Soc., 118, 1(1993), 190-196.
Author:
Daniel Breaz
Department of Mathematics
”1 Decembrie 1918” University
Alba Iulia
Romania
e-mail:dbreaz@uab.ro

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