Ann. Funct. Anal. 3 (2012), no. 2, 183–190
A nnals of F unctional A nalysis
ISSN: 2008-8752 (electronic)
URL: www.emis.de/journals/AFA/
ON THE CONVEXITY OF CERTAIN INTEGRAL OPERATORS
VASILE MARIUS MACARIE1∗ AND DANIEL BREAZ2
Communicated by K. Gurlebeck
Abstract. In this paper we consider the classes of starlike functions of order
α, convex functions of order α and we study the convexity and α-order convexity for some general integral operators. Several corollaries of the main results
are also considered.
1. Introduction and preliminaries
Let U = {z ∈ C : |z| < 1} be the unit disk 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 }.
Consider S ∗ the class of starlike functions in unit disk, defined by
0 zf (z)
∗
S = f ∈ A : Re
> 0, z ∈ U .
f (z)
Definition 1.1. A function f ∈ S is a starlike function of order α, 0 ≤ α < 1
and denote this class by S ∗ (α) if f verifies the inequality
0 zf (z)
> α, (z ∈ U ).
Re
f (z)
Denote with K the class of convex functions in U , defined by
00
zf (z)
K = f ∈ A : Re
+ 1 > 0, z ∈ U .
f 0 (z)
Date: Received: 30 January 2012; Accepted: 14 June 2012.
∗
Corresponding author.
2010 Mathematics Subject Classification. Primary 30C45; Secondary 30C55, 47G10.
Key words and phrases. Analytic function, integral operator, open unit disk, convexity, starlike function.
183
184
V.M. MACARIE, D. BREAZ
Definition 1.2. A function f ∈ S is convex function of order α, 0 ≤ α < 1 and
denote this class by K(α) if f verifies the inequality
00
zf (z)
Re
+ 1 > α, (z ∈ U ).
f 0 (z)
Definition 1.3. A function f ∈ A is said to be in the class R(α) if Re(f 0 (z)) > α,
(z ∈ U ).
It is well known that K(α) ⊂ S ∗ (α) ⊂ S.
Frasin and Jahangiri defined in [1] the family B(µ, α), µ ≥ 0, 0 ≤ α < 1 so
that it consists of functions f ∈ A satisfying the condition
µ
0
z
f (z)
< 1 − α, (z ∈ U ).
−
1
(1.1)
f (z)
For µ = 0 we have B(0, α) ≡ R(α) and for µ = 1 we have B(1, α) ≡ S ∗ (α).
In this paper we will obtain the order of convexity of the following integral
operators:
Z zY
n
γi
Hn (z) =
tefi (t) dt,
(1.2)
0
Z
H(z) =
i=1
n zY
0
i=1
fi (t)
t
β1
tn(β−1) dt,
n
zY
Z
(fi (t))β−1 dt,
G(z) =
0
(1.3)
(1.4)
i=1
where the functions fi (t) are in B(µi , αi ) for all i = 1, 2, · · · , n.
Lemma 1.4. [2](General Schwarz lemma). Let the function f be regular in the
disk UR = {z ∈ C : |z| < R}, with |f (z)| < M for fixed M . If f has one zero
with multiplicity order bigger than m for z = 0, then
|f (z)| ≤
M
· |z|m
Rm
(z ∈ UR ).
The equality can hold only if
f (z) = eiθ ·
M
· zm,
Rm
where θ is constant.
2. Main results
Theorem 2.1. Let fi (z) ∈ A be in the class B(µi , αi ), µi ≥ 0, 0 ≤ αi < 1 for
all i = 1, 2, · · · , n. If |fi (z)| ≤ Mi (Mi ≥ 1, z ∈ U ) for all i = 1, 2, · · · , n, then
the integral operator
Z zY
n
γi
Hn (z) =
tefi (t) dt
0
i=1
ON THE CONVEXITY OF CERTAIN INTEGRAL OPERATORS
185
is in K(δ), where
δ =1−
n
X
|γi | · [1 + (2 − αi )Miµi ]
i=1
and
n
P
|γi | · [1 + (2 − αi )Miµi ] < 1, γi ∈ C for all i = 1, 2, · · · n..
i=1
Proof. Let fi ∈ A be in the class B(µi , αi ), µi ≥ 0, 0 ≤ αi < 1. It follows from
(1.2) that
n
n
n
n
Z z P
P
P
P
γi
γi fi (t)
γi
γi fi (z)
0
.
ti=1 ei=1
Hn (z) =
dt and Hn (z) = z i=1 ei=1
0
Also
n
P
Hn00 (z) = z i=1
γi −1
n
P
· ei=1
γi fi (z)
"
n
X
i=1
Then
Hn00 (z)
Hn0 (z)
n
P
=
γi + z
i=1
n
P
γi + z
n
X
#
γi fi0 (z)
i=1
γi fi0 (z)
i=1
z
and, hence
00 n
n
n
n
X
X
X
X
zHn (z) 0
= |γi | + |z|
|γi | · |fi0 (z)|
γi + z
γi fi (z) ≤
H 0 (z) n
i=1
i=1
i=1
i=1
µi n
n
X
X
0
fi (z) µi z
(2.1)
≤
|γi | + |z| ·
|γi | · fi (z)
·
f
(z)
z
i
i=1
i=1
fi (z) ≤ Mi , for all i = 1, 2, · · · , n.
Applying the General Schwarz lemma, we have z Therefore, from (2.1), we obtain
00 X
µi n
n
X
zHn (z) 0
z
≤
fi (z)
· M µi , (z ∈ U ).
|γ
|
+
|γ
|
·
(2.2)
i
i
i
H 0 (z) fi (z)
n
i=1
i=1
From (1.1) and (2.2), we see that
00 n
zHn (z) X
≤
|γi | · [1 + (2 − αi )Miµi ] = 1 − δ.
H 0 (z) n
i=1
This completes the proof.
Letting Mi = M for all i = 1, 2, · · · , n in Theorem 2.1, we have
Corollary 2.2. Let fi (z) ∈ A be in the class B(µi , αi ), µi ≥ 0, 0 ≤ αi < 1 for
all i = 1, 2, · · · , n. If |fi (z)| ≤ M (M ≥ 1, z ∈ U ) for all i = 1, 2, · · · , n, then
the integral operator
Z zY
n
γi
Hn (z) =
tefi (t) dt
0
i=1
186
V.M. MACARIE, D. BREAZ
is in K(δ), where
δ =1−
n
X
|γi | · [(2 − αi )M µi + 1]
i=1
and
n
P
|γi | · [(2 − αi )M µi + 1] < 1, γi ∈ C for all i = 1, 2, · · · n..
i=1
Letting µi = 0 for all i = 1, 2, · · · , n in Corollary 2.2, we have
Corollary 2.3. Let fi (z) ∈ A be in the class R(αi ), 0 ≤ αi < 1 for all i =
1, 2, · · · , n. Then the integral operator defined in (1.2) is in K(δ), where
δ =1−
n
X
|γi |(3 − αi )
i=1
and
n
P
|γi |(3 − αi ) < 1, γi ∈ C for all i = 1, 2, · · · , n.
i=1
Letting µi = 1 for all i = 1, 2, · · · , n in Corollary 2.2, we have
Corollary 2.4. Let fi ∈ A be in the class S ∗ (αi ), 0 ≤ αi < 1 for all i =
1, 2, · · · , n. If |fi (z)| ≤ M (M ≥ 1, z ∈ U ) for all i = 1, 2, · · · , n, then the
integral operator defined in (1.2) is in K(δ), where
δ =1−
n
X
|γi | · [1 + (2 − αi )M ]
i=1
and
n
P
|γi | · [1 + (2 − αi )M ] < 1, γi ∈ C for all i = 1, 2, · · · , n.
i=1
Letting αi = δ = 0 for all i = 1, 2, · · · , n in Corollary 2.4, we have
Corollary 2.5. Let fi ∈ A be starlike functions in U for all i = 1, 2, · · · , n. If
|fi (z)| ≤ M (M ≥ 1, z ∈ U ) for all i = 1, 2, · · · , n then the integral operator
n
P
1
defined in (1.2) is convex in U , where
|γi | =
, γi ∈ C for all i =
2M + 1
i=1
1, 2, · · · , n.
Theorem 2.6. Let fi (z) be in the class B(µi , αi ), µi ≥ 1, 0 ≤ αi < 1 for all
i = 1, 2, · · · , n. If |fi (z)| ≤ Mi (Mi ≥ 1, z ∈ U ) for all i = 1, 2, · · · , n, then the
integral operator
1
Z zY
n fi (t) β n(β−1)
H(z) =
t
dt,
t
0 i=1
is in K(δ), where
(
δ =1−
and
)
n
1 X
(2 − αi )Miµi −1 + 1 + n|β − 1|
|β| i=1
n 1 P
(2 − αi )Miµi −1 + 1 + n|β − 1| < 1, β ∈ C \ {0}.
|β| i=1
ON THE CONVEXITY OF CERTAIN INTEGRAL OPERATORS
187
Proof. Let fi (z) be in the class B(µi , αi ), µi ≥ 1, 0 ≤ αi < 1, for all i =
1, 2, · · · , n. It follows from (1.3) that
H 00 (z)
1 f10 (z) · z − f1 (z)
1 fn0 (z) · z − fn (z) n(β − 1)
=
·
+
·
·
·
+
·
+
H 0 (z)
β
zf1 (z)
β
zfn (z)
z
and
n zH 00 (z)
1 X fi0 (z) · z
= ·
− 1 + n(β − 1).
H 0 (z)
β i=1
fi (z)
So that
00 n X
zH (z) zfi0 (z) ≤ 1 ·
+ 1 + n|β − 1|
H 0 (z) |β|
fi (z) i=1
!
µi n
fi (z) µi −1 1 X 0
z
·
·
f (z)
(2.3)
≤
+ 1 + n|β − 1|.
|β| i=1 i
fi (z)
z
fi (z) ≤ Mi , (z ∈ U ) for all
Applying the General Schwarz lemma, we have z i = 1, 2, · · · , n. Therefore, from (2.3), we obtain
00 µi n X
zH (z) 0
1
z
µi −1
≤
+ 1 + n|β − 1|, (z ∈ U ). (2.4)
· Mi
H 0 (z) |β| ·
fi (z) fi (z)
i=1
From (1.1) and (2.4), we see that
00 n
X
zH (z) ≤ 1
(2 − αi )Miµi −1 + 1 + n|β − 1| = 1 − δ.
H 0 (z) |β|
i=1
This completes the proof.
Letting Mi = M for all i = 1, 2, · · · , n in Theorem 2.6, we have
Corollary 2.7. Let fi (z) be in the class B(µi , αi ), µi ≥ 1, 0 ≤ αi < 1 for all
i = 1, 2, · · · , n. If |fi (z)| ≤ M (M ≥ 1, z ∈ U ) for all i = 1, 2, · · · , n, then the
integral operator
1
Z zY
n fi (t) β n(β−1)
H(z) =
t
dt
t
0 i=1
is in K(δ), where
(
δ =1−
and
)
n
1 X
(2 − αi )M µi −1 + 1 + n|β − 1|
|β| i=1
n
1 P
[(2 − αi )M µi −1 + 1] + n|β − 1| < 1, β ∈ C \ {0}.
|β| i=1
Letting δ = 0 in Corollary 2.7 we have
188
V.M. MACARIE, D. BREAZ
Corollary 2.8. Let fi (z) be in the class B(µi , αi ), µi ≥ 1, 0 ≤ αi < 1 for all
i = 1, 2, · · · , n. If |fi (z)| ≤ M (M ≥ 1, z ∈ U ) for all i = 1, 2, · · · , n, then the
integral operator defined in (1.3) is convex function in U , where
n
1 X
(2 − αi )M µi −1 + 1 + n|β − 1| = 1, β ∈ C \ {0}.
|β| i=1
Letting µi = 1 for all i = 1, 2, · · · , n in Corollary 2.7, we have
Corollary 2.9. Let fi (z) be in the class S ∗ (αi ), 0 ≤ αi < 1 for all i = 1, 2, · · · , n.
If |fi (z)| ≤ M (M ≥ 1, z ∈ U ) for all i = 1, 2, · · · , n, then the integral operator
defined in (1.3) is in K(δ), where
"
#
n
1 X
δ =1−
(3 − αi ) + n|β − 1|
|β| i=1
n
1 P
(3 − αi ) + n|β − 1| < 1 for all β ∈ C \ {0}.
|β| i=1
Letting n = 1 and αi = δ = 0 for all i = 1, 2, · · · , n in Corollary 2.9, we have
and
Corollary 2.10. Let f (z) be a starlike function in U . If |f (z)| ≤ M (M ≥
R z f (t) β1 β−1
t dt is convex in U , where
1, z ∈ U ) then the integral operator 0
t
3
+ |β − 1| = 1, β ∈ C \ {0}.
|β|
Theorem 2.11. Let fi (z) be in the class B(µi , αi ), µi ≥ 1, 0 ≤ αi < 1 for all
i = 1, 2, · · · , n. If |fi (z)| ≤ Mi (Mi ≥ 1, z ∈ U ), for all i = 1, 2, · · · , n then the
integral operator
Z zY
n
G(z) =
(fi (t))β−1 dt
0
i=1
is in K(δ), where
n
X
δ = 1 − |β − 1| ·
(2 − αi )Miµi −1
i=1
and |β − 1| ·
n
P
(2 − αi )Miµi −1 < 1, β ∈ C.
i=1
Proof. Let fi (z) be in the class B(µi , αi ), µi ≥ 1, 0 ≤ αi < 1 for all i = 1, 2, · · · , n.
It follows from (1.4) that
n
X
G00 (z)
fi0 (z)
=
(β
−
1)
G0 (z)
f (z)
i=1 i
and, hence
00 !
n X
zfi0 (z) zG (z) G0 (z) ≤ |β − 1|
fi (z) i=1
!
µi n X
0
fi (z) µi −1 z
fi (z)
·
≤ |β − 1|
f
(z)
z
i
i=1
(2.5)
ON THE CONVEXITY OF CERTAIN INTEGRAL OPERATORS
189
fi (z) ≤ Mi , (z ∈ U ) for all
Applying the General Schwarz lemma, we have z i = 1, 2, · · · , n. Therefore, from (2.5), we obtain
!
00 µi n X
zG (z) 0
z
· M µi −1 , z ∈ U.
fi (z)
(2.6)
i
G0 (z) ≤ |β − 1|
f
(z)
i
i=1
From (1.1) and (2.6), we see that
00 n
X
zG (z) ≤ |β − 1| ·
(2 − αi )Miµi −1 = 1 − δ.
G0 (z) i=1
This completes the proof.
Letting Mi = M for all i = 1, 2, · · · , n in Theorem 2.11, we have
Corollary 2.12. Let fi (z) be in the class B(µi , αi ), µi ≥ 1, 0 ≤ αi < 1 for all
i = 1, 2, · · · , n. If |fi (z)| ≤ M (M ≥ 1, z ∈ U ) then the integral operator
Z zY
n
(fi (t))β−1 dt
G(z) =
0
i=1
is in K(δ), where
n
X
δ = 1 − |β − 1|
(2 − αi )M µi −1
i=1
and |β − 1|
n
P
(2 − αi )M µi −1 < 1, β ∈ C.
i=1
Letting δ = 0 in Corollary 2.12 we have
Corollary 2.13. Let fi (z) be in the class B(µi , αi ), µi ≥ 1, 0 ≤ αi < 1 for all
i = 1, 2, · · · , n. If |fi (z)| ≤ M (M ≥ 1, z ∈ U ) then the integral operator defined
in (1.4) is convex function in U , where
|β − 1| = P
n
1
(2 − αi
, β ∈ C.
)M µi −1
i=1
Letting µi = 1 for all i = 1, 2, · · · , n in Corollary 2.12, we have
Corollary 2.14. Let fi (z) be in the class S ∗ (αi ), 0 ≤ αi < 1 for all i =
1, 2, · · · , n. If |fi (z)| ≤ M (M ≥ 1, z ∈ U ) then the integral operator defined in
(1.4) is in K(δ), where
n
X
δ = 1 − |β − 1|
(2 − αi )
i=1
and |β − 1|
n
P
(2 − αi ) < 1, β ∈ C.
i=1
Letting n = 1 and αi = δ = 0 for all i = 1, 2, · · · , n in Corollary 2.14, we have
190
V.M. MACARIE, D. BREAZ
Corollary 2.15. Let f (z) be a starlike
R z function in U . If |f (z)| ≤ M (M ≥
1, z ∈ U ) then the integral operator 0 f (t)β−1 dt is convex in U , where |β − 1| =
1
, β ∈ C.
2
References
1. B.A. Frasin and J. Jahangiri, A new and comprehensive class of analytic functions, Anal.
Univ. Oradea, Fasc. Math. 15 (2008), 59–62.
2. Z. Nehari, Conformal Mapping, Dover, New York, NY, USA, 1975.
1
Department of Mathematics, University of Pitesti, Targu din Vale Str., No.
1, 110040, Pitesti, Arges, Romania.
E-mail address: macariem@yahoo.com
2
Department of Mathematics, ”1 Decembrie 1918” University of Alba Iulia,
N. Iorga Str., No. 11-13, 510000, Alba Iulia, Romania.
E-mail address: dbreaz@uab.ro