Journal of Inequalities in Pure and
Applied Mathematics
SOME CONVEXITY PROPERTIES FOR A GENERAL INTEGRAL
OPERATOR
volume 7, issue 5, article 177,
2006.
DANIEL BREAZ AND NICOLETA BREAZ
Department of Mathematics
" 1 Decembrie 1918 " University
Alba Iulia
Romania.
EMail: dbreaz@uab.ro
EMail: nbreaz@uab.ro
Received 12 September, 2006;
accepted 18 October, 2006.
Communicated by: G. Kohr
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
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Abstract
In this paper we consider the classes of starlike functions, starlike functions of
order α, convex functions, convex functions of order α and the classes of the
univalent functions denoted by SH (β), SP and SP (α, β). On these classes
we study the convexity and α- order convexity for a general integral operator.
Some Convexity Properties for
a General Integral Operator
2000 Mathematics Subject Classification: 30C45.
Key words: Univalent function, Integral operator, Convex function, Analytic function,
Starlike function.
Daniel Breaz and Nicoleta Breaz
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
6
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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
A = f ∈ H (U ) , f (z) = z + a2 z 2 + a3 z 3 + · · · , z ∈ U
the class of analytic functions in U and S = {f ∈ A : f is univalent in U }.
We denote by S ∗ the class of starlike functions that are defined as holomorphic
functions in the unit disc with the properties f (0) = f 0 (0) − 1 = 0 and
Re
zf 0 (z)
> 0,
f (z)
z ∈ U.
A function f ∈ A is a starlike function by the order α, 0 ≤ α < 1 if f satisfies
the inequality
zf 0 (z)
Re
> α, z ∈ U.
f (z)
∗
We denote this class by S (α). Also, we denote by K the class of convex
functions that are defined as holomorphic functions in the unit disc with the
properties f (0) = f 0 (0) − 1 = 0 and
00
zf (z)
Re
+ 1 > 0, z ∈ U.
f 0 (z)
A function f ∈ A is a convex function by the order α, 0 ≤ α < 1 if f verifies
the inequality
00
zf (z)
+ 1 > α, z ∈ U.
Re
f 0 (z)
Some Convexity Properties for
a General Integral Operator
Daniel Breaz and Nicoleta Breaz
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We denote this class by K (α).
In the paper [5] J. Stankiewicz and A. Wisniowska introduced the class of
univalent functions, SH (β), β > 0 defined by:
(1.1)
0
√
zf (z)
2 − 1 f (z) − 2β
√
√ zf 0 (z)
2−1 ,
< Re
2
+ 2β
f (z)
f ∈ S,
for all z ∈ U .
Also, in the paper [3] F. Ronning introduced the class of univalent functions,
SP , defined by
zf 0 (z) zf 0 (z)
(1.2)
Re
>
− 1 , f ∈ S,
f (z)
f (z)
for all z ∈ U . The geometric interpretation of the relation (1.2) is that the class
SP is the class of all functions f ∈ S for which the expression zf 0 (z) /f (z) ,
z ∈ U takes all values in the parabolic region
Ω = {ω : |ω − 1| ≤ Re ω} = ω = u + iv : v 2 ≤ 2u − 1 .
In the paper [3] F. Ronning introduced the class of univalent functions SP (α, β) ,
α > 0, β ∈ [0, 1), as the class of all functions f ∈ S which have the property:
0
0
zf (z)
≤ Re zf (z) + α − β,
(1.3)
−
(α
+
β)
f (z)
f (z)
Some Convexity Properties for
a General Integral Operator
Daniel Breaz and Nicoleta Breaz
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for all z ∈ U . Geometric interpretation: f ∈ SP (α, β) if and only if zf 0 (z) /f (z) ,
z ∈ U takes all values in the parabolic region
Ωα,β = {ω : |ω − (α + β)| ≤ Re ω + α − β}
= ω = u + iv : v 2 ≤ 4α (u − β) .
We consider the integral operator Fn , defined by:
α
α
Z z
fn (t) n
f1 (t) 1
dt
(1.4)
Fn (z) =
· ··· ·
t
t
0
and we study its properties.
Remark 1. We observe that for
= 1 and α1 = 1 we obtain the integral
R znf (t)
operator of Alexander, F (z) = 0 t dt.
Some Convexity Properties for
a General Integral Operator
Daniel Breaz and Nicoleta Breaz
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2.
Main Results
Theorem 2.1. Let αi , i ∈ {1, . . . , n} be real numbers with the properties αi > 0
for i ∈ {1, . . . , n} and
n
X
αi ≤ n + 1.
i=1
We suppose that the functions fi , i = {1,. . ., n} are the starlike functions by
order α1i , i ∈ {1, . . . , n}, that is fi ∈ S ∗ α1i for all i ∈ {1, . . . , n} . In these
conditions the integral operator defined in (1.4) is convex.
Proof. We calculate for Fn the derivatives of the first and second order. From
(1.4) we obtain:
α
α
fn (z) n
f1 (z) 1
0
Fn (z) =
· ··· ·
z
z
(z) =
n
X
αi
i=1
fi (z)
z
αi −1 zfi0 (z) − fi (z)
zfi (z)
Y
n j=1
fj (z)
z
α j
.
j6=i
Fn00 (z)
Fn0 (z)
(2.1)
= α1
zf10
Fn00 (z)
= α1
Fn0 (z)
(z) − f1 (z)
zf1 (z)
f10 (z) 1
−
f1 (z) z
+ · · · + αn
+ · · · + αn
zfn0
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(z) − fn (z)
zfn (z)
fn0 (z) 1
−
fn (z) z
Daniel Breaz and Nicoleta Breaz
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and
Fn00
Some Convexity Properties for
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.
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By multiplying the relation (2.1) with z we obtain:
0
X
n
n
zFn00 (z) X
zfi (z)
zfi0 (z)
(2.2)
α
−
1
=
α
− α1 − · · · − αn .
=
i
i
Fn0 (z)
f
(z)
f
(z)
i
i
i=1
i=1
The relation (2.2) is equivalent with
zFn00 (z)
zf10 (z)
zfn0 (z)
(2.3)
+
1
=
α
+
·
·
·
+
α
− α1 − · · · − αn + 1.
1
n
Fn0 (z)
f1 (z)
fn (z)
From (2.3) we obtain that:
00
zFn (z)
(2.4)
Re
+1
Fn0 (z)
zf 0 (z)
zf 0 (z)
+ · · · + αn Re n
− α1 − · · · − αn + 1.
= α1 Re 1
f1 (z)
fn (z)
zf 0 (z)
∗
But fi ∈ S α1i , for all i ∈ {1, . . . , n} , so Re fii(z) > α1i , for all i ∈
{1, . . . , n} . We apply this affirmation in the equality (2.4) and obtain:
00
zFn (z)
1
1
(2.5)
Re
+ 1 > α1 + · · · + αn
− α1 − · · · − αn + 1
0
Fn (z)
α1
αn
n
X
=n+1−
αi .
i=1
But, in accordance with the hypothesis, we obtain:
00
zFn (z)
Re
+1 >0
Fn0 (z)
so, Fn is a convex function.
Some Convexity Properties for
a General Integral Operator
Daniel Breaz and Nicoleta Breaz
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Theorem 2.2. Let αi , i ∈ {1, . . . , n}, be real numbers with the properties αi >
0 for i ∈ {1, . . . , n} and
n
X
αi ≤ 1.
i=1
We suppose that the functions fi , i = {1, . . . , n}, are the starlike
P functions.
Then the integral operator defined in (1.4) is convex by order, 1 − ni=1 αi .
Proof. Following the same steps as in Theorem 2.1, we obtain:
0
X
n
n
zFn00 (z) X
zfi (z)
zfi0 (z)
(2.6)
=
α
−
1
=
α
− α1 − · · · − αn .
i
i
Fn0 (z)
f
(z)
f
(z)
i
i
i=1
i=1
Some Convexity Properties for
a General Integral Operator
Daniel Breaz and Nicoleta Breaz
The relation (2.6) is equivalent with
(2.7)
zFn00 (z)
zf10 (z)
zfn0 (z)
+
1
=
α
+
·
·
·
+
α
− α1 − · · · − αn + 1.
1
n
Fn0 (z)
f1 (z)
fn (z)
From (2.7) we obtain that:
00
zFn (z)
(2.8)
Re
+1
Fn0 (z)
zf 0 (z)
zf 0 (z)
= α1 Re 1
+ · · · + αn Re n
− α1 − · · · − αn + 1.
f1 (z)
fn (z)
zf 0 (z)
But fi ∈ S ∗ for all i ∈ {1, . . . , n} , so Re fii(z) > 0 for all i ∈ {1, . . . , n} . We
apply this affirmation in the equality (2.8) and obtain that:
00
n
X
zFn (z)
+ 1 > α1 ·0+· · ·+αn ·0−α1 −· · ·−αn +1 = 1−
αi .
(2.9) Re
Fn0 (z)
i=1
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But in accordance with the inequality (2.9), obtain that
Re
n
X
zFn00 (z)
+1 >1−
αi
Fn0 (z)
i=1
so, Fn is a convex function by order 1 −
Pn
i=1
αi .
Theorem 2.3. Let αi , i ∈ {1, . . . , n}, be real numbers with the properties αi >
0, for i ∈ {1, . . . , n} and
√
n
X
2
√
√ .
(2.10)
αi ≤
2β
2
−
1
+ 2
i=1
We suppose that fi ∈ SH (β), for i = {1, . . . , n} and β > 0. In these conditions, the integral operator defined in (1.4) is convex.
Proof. Following the same steps as in Theorem 2.1, we obtain that:
n
(2.11)
We multiply the relation (2.11) with
(2.12)
n
X zf 0 (z) X
zFn00 (z)
+
1
=
αi i
−
αi + 1.
0
Fn (z)
fi (z)
i=1
i=1
√
2
zFn00 (z)
Fn0 (z)
√
Some Convexity Properties for
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Daniel Breaz and Nicoleta Breaz
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2 and obtain:
X
n
n
√ zfi0 (z) √ X
√
+1 =
2αi
− 2
αi + 2.
fi (z)
i=1
i=1
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The equality (2.12) is equivalent with:
X
n √
√ zFn00 (z)
√ zfi0 (z)
2
+1 =
αi 2
+ 2αi β
2−1
Fn0 (z)
fi (z)
i=1
n
n
√ X
√
X
√
2−1 − 2
αi + 2.
−
2αi β
i=1
i=1
We calculate the real part from both terms of the above equality and obtain:
00
X
n √
√
√ zfi0 (z)
zFn (z)
2 Re
+1 =
αi Re
2
+ 2β
2−1
Fn0 (z)
fi (z)
i=1
n
n
√
√ X
X
√
2αi β
2−1 − 2
αi + 2.
−
i=1
i=1
Because fi ∈ SH (β) for i = {1, . . . , n}, we apply in the above relation the
inequality (1.1) and obtain:
0
00
X
n
√
√
zfi (z)
zFn (z)
2 Re
+1 >
αi − 2β
2 − 1 Fn0 (z)
f
(z)
i
i=1
n
n
√
√ X
X
√
−
2αi β
2−1 − 2
αi + 2.
i=1
i=1
0
√
zf (z)
Because αi fii(z) − 2β
2 − 1 > 0, for all i ∈ {1, . . . , n}, we obtain that
00
n
n
√
√ X
X
√
√
zFn (z)
+
1
>
−
(2.13)
2 Re
2α
β
2
−
1
−
2
α
+
2.
i
i
Fn0 (z)
i=1
i=1
Some Convexity Properties for
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Daniel Breaz and Nicoleta Breaz
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Using the hypothesis (2.10), we have:
00
zFn (z)
(2.14)
Re
+ 1 > 0,
Fn0 (z)
so, Fn is a convex function.
√
Corollary 2.4. Let α be real numbers with the properties 0 < α ≤
2
√ ,
2β ( 2−1)+ 2
√
β > 0. We suppose that thefunctions
f ∈ SH (β). In these conditions the inteR z f (t) α
gral operator, F (z) = 0
dt is convex.
t
Proof. In Theorem 2.3, we consider n = 1, α1 = α and f1 = f .
Theorem 2.5. Let αi , i ∈ {1, . . . , n} be real numbers with the properties αi > 0
for i ∈ {1, . . . , n},
Some Convexity Properties for
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Daniel Breaz and Nicoleta Breaz
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n
X
(2.15)
JJ
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αi < 1
i=1
P
and 1 − ni=1 αi ∈ [0, 1). We consider the functions fi , fi ∈ SP for i =
{1, . . . ,P
n}. In these conditions, the integral operator defined in (1.4) is convex
by 1 − ni=1 αi order.
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Proof. Following the same steps as in Theorem 2.1, we have:
(2.16)
zFn00 (z)
Fn0 (z)
+1=
n
X
i=1
αi
zfi0
(z)
−
fi (z)
n
X
i=1
αi + 1.
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We calculate the real part from both terms of the above equality and obtain:
(2.17)
Re
0
X
X
n
n
zFn00 (z)
zfi (z)
−
+1 =
αi Re
αi + 1.
Fn0 (z)
fi (z)
i=1
i=1
Because fi ∈ SP for i = {1, . . . , n} we apply in the above relation the inequality (1.2) and obtain:
0
00
X
n
n
zfi (z)
X
zFn (z)
(2.18)
Re
+
1
>
α
−
1
−
αi + 1.
i
Fn0 (z)
f
(z)
i
i=1
i=1
Some Convexity Properties for
a General Integral Operator
Daniel Breaz and Nicoleta Breaz
0
zfi (z)
Because αi fi (z) − 1 > 0, for all i ∈ {1, . . . , n}, we get
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(2.19)
Re
zFn00 (z)
Fn0 (z)
n
X
+1 >1−
αi .
i=1
Using the hypothesis, we obtain that Fn is a convex function by 1 −
order.
P
Remark 2. If ni=1 αi = 1 then
00
zFn (z)
+ 1 > 0,
(2.20)
Re
Fn0 (z)
so, Fn is a convex function.
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Pn
i=1
αi
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Corollary 2.6. Let γ be a real number with the property 0 < γ < 1.We suppose
γ
Rz
that f ∈ SP . In these conditions the integral operator F (z) = 0 f (t)
dt is
t
convex of 1 − γ order.
Proof. In Theorem 2.5, we consider n = 1, α1 = γ and f1 = f .
Theorem 2.7. We suppose that f ∈ SP . In this condition, the integral operator
of Alexander, defined by
Z z
f (t)
(2.21)
F1 (z) =
dt,
t
0
Some Convexity Properties for
a General Integral Operator
Daniel Breaz and Nicoleta Breaz
is convex.
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Proof. We have:
(2.22)
Re
zF100 (z)
zf 0 (z) zf 0 (z)
> 0.
+
1
=
Re
>
−
1
f (z)
F10 (z)
f (z)
So, the relation (2.22) implies that the Alexander operator is convex.
Remark 3. Theorem 2.7 can be obtained from Corollary 2.6, for γ = 1.
Theorem 2.8. Let αi , i ∈ {1, . . . , n} be real numbers with the properties αi > 0
for i ∈ {1, . . . , n},
(2.23)
n
X
i=1
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1
αi <
,
α−β+1
α > 0, β ∈ [0, 1)
J. Ineq. Pure and Appl. Math. 7(5) Art. 177, 2006
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P
and (β − α − 1) ni=1 αi + 1 ∈ (0, 1). We suppose that fi ∈ SP (α, β), for
i = {1, . . . , n}. In these
P conditions, the integral operator defined in (1.4) is
convex by (β − α − 1) ni=1 αi + 1 order.
Proof. Following the same steps as in Theorem 2.1, we obtain that:
0
n
n
X
zFn00 (z) X
zfi (z)
(2.24)
=
αi
+ α − β + (β − α − 1)
αi .
Fn0 (z)
fi (z)
i=1
i=1
and
n
X
zFn00 (z)
(2.25)
+
1
=
αi
Fn0 (z)
i=1
zfi0 (z)
+α−β
fi (z)
+ (β − α − 1)
n
X
Some Convexity Properties for
a General Integral Operator
αi + 1.
i=1
We calculate the real part from both terms of the above equality and get:
( n
00
)
X zf 0 (z)
zFn (z)
i
(2.26) Re
+ 1 = Re
αi
+α−β
Fn0 (z)
f
(z)
i
i=1
+ (β − α − 1)
n
X
αi + 1.
Daniel Breaz and Nicoleta Breaz
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i=1
Because fi ∈ SP (α, β) for i = {1, . . . , n} we apply in the above relation the
inequality (1.3) and obtain:
00
zFn (z)
+1
(2.27) Re
Fn0 (z)
0
n
n
X
X
zfi (z)
≥
αi − (α + β) + (β − α − 1)
αi + 1.
fi (z)
i=1
i=1
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0
zf (z)
Since αi fii(z) − (α + β) > 0, for all i ∈ {1, . . . , n}, using the inequality
(1.3), we have
00
n
X
zFn (z)
(2.28)
Re
+ 1 ≥ (β − α − 1)
αi + 1 > 0.
Fn0 (z)
i=1
P
From (2.28), since (β − α − 1) ni=1 αi + 1 ∈ (0, 1),
obtain that the integral
Pwe
n
operator defined in (1.4) is convex by (β − α − 1) i=1 αi + 1 order.
1
Corollary 2.9. Let γ be a real number with the property 0 < γ < α−β+1
,
α > 0, β ∈ [0, 1). We suppose
that f ∈ SP (α, β). In these conditions, the
R z f (t) γ
dt is convex.
integral operator F (z) = 0
t
Proof. In Theorem 2.8, we consider n = 1, α1 = γ and f1 = f .
For α = β ∈ (0, 1) we obtain the class S (α, α) that is characterized by the
property
0
0
zf (z)
≤ Re zf (z) .
(2.29)
−
2α
f (z)
f (z)
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Corollary 2.10. Let αi , i ∈ {1, . . . , n} be real numbers with the properties
αi > 0 for i ∈ {1, . . . , n} and
(2.30)
1−
n
X
i=1
αi ∈ [0, 1) .
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We consider the functions fi , fi ∈ SP (α, α), i = {1, . . . , n}, α ∈ (0,
In
P1).
n
these conditions, the integral operator defined in (1.4) is convex by 1 − i=1 αi
order.
Proof. From (1.4) we obtain
n
n
zFn00 (z) X zfi0 (z) X
=
αi
−
αi ,
Fn0 (z)
fi (z)
i=1
i=1
(2.31)
which is equivalent with
X
00
n
n
zFn (z)
zfi0 (z) X
(2.32)
Re
+1 =
αi Re
−
αi + 1.
Fn0 (z)
fi (z)
i=1
i=1
Some Convexity Properties for
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Daniel Breaz and Nicoleta Breaz
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From (2.31) and (2.32), we have:
0
00
X
n
n
X
zfi (z)
zFn (z)
+1 >
αi − 2α + 1 −
αi .
(2.33)
Re
Fn0 (z)
fi (z)
i=1
i=1
Since
0
zfi (z)
α
−
2α
> 0, for all i ∈ {1, . . . , n}, from (2.33), we get:
i=1 i fi (z)
Pn
(2.34)
Re
n
X
zFn00 (z)
+1 >1−
αi .
Fn0 (z)
i=1
Now,
Pn from (2.34) we obtain that the operator defined in (1.4) is convex by 1 −
i=1 αi order.
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References
[1] M. ACU, Close to convex functions associated with some hyperbola, Libertas Mathematica, XXV (2005), 109–114.
[2] D. BREAZ AND N. BREAZ, Two integral operators, Studia Universitatis
Babeş -Bolyai, Mathematica, Cluj Napoca, (3) (2002), 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.
[5] J. STANKIEWICZ AND A. WISNIOWSKA, Starlike functions associated
with some hyperbola, Folia Scientiarum Universitatis Tehnicae Resoviensis
147, Matematyka, 19 (1996), 117–126.
Some Convexity Properties for
a General Integral Operator
Daniel Breaz and Nicoleta Breaz
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