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On univalence of two integral operators
Applied Mathematics Letters 23 (2010) 1407–1411
Contents lists available at ScienceDirect
Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
On univalence of two integral operators Virgil Pescar a , Nicoleta Breaz b,∗ a
Department of Mathematics, ‘‘Transilvania’’ University of Braşov, Faculty of Mathematics and Computer Science, 500091 Braşov, Romania
b
Department of Mathematics, ‘‘1 Decembrie 1918’’ University of Alba Iulia, Faculty of Science, 510009 Alba Iulia, Romania
article
abstract
info
Article history: Received 26 April 2010 Accepted 21 July 2010
In this work we consider two integral operators. The integral operators were constructed on the basis of the fact that the number of functions from the composition of the operators is the entire part of a complex number modulus. © 2010 Elsevier Ltd. All rights reserved.
Keywords: Integral operator Univalence Integer part Modulus
1. Introduction Let us consider the open unit disk U and let A be the class of analytic functions defined by f (z ) = z + a2 z 2 + · · ·. We denote by S the class of univalent functions. Theorem 1.1 ([1]). If the function f belongs to the class S , then for any complex number γ , |γ | ≤ Fγ (z ) =
Z z
f (t ) t
0
1 , 4
the function
γ dt
is in the class S . Theorem 1.2 ([2]). If the function f is regular in the open unit disk U, f (z ) = z + a2 z 2 + · · · and
zf 00 (z ) ≤ 1, 1 − |z |2 0 f (z )
for all z ∈ U, then the function f is univalent in U. Theorem 1.3 ([3]). Let α be a complex number, Re α > 0 and f (z ) = z + a2 z 2 + · · · be a regular function in U. If 1 − |z |2Re α zf 00 (z )
f 0 (z ) ≤ 1,
Re α
for all z ∈ U, then for any complex number β , Re β ≥ Re α , the function z
Z
Fβ (z ) = β
t β−1 f 0 (t )dt
β1
0
is in the class S .
∗
Corresponding author. Fax: +40 258812630. E-mail addresses: [email protected] (V. Pescar), [email protected], [email protected] (N. Breaz).
0893-9659/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2010.07.007
1408
V. Pescar, N. Breaz / Applied Mathematics Letters 23 (2010) 1407–1411
Theorem 1.4 ([4]). If the function g is regular in U and |g (z )| < 1 in U, then for all ξ ∈ U, the following inequalities hold:
g (ξ ) − g (z ) ξ − z ≤ 1 − g (z )g (ξ ) 1 − z ξ
(1.1)
and
0 1 − |g (z )|2 g (z ) ≤ , 1 − |z |2 u and the equalities hold in the case g (z ) = 1z++uz , where || = 1 and |u| < 1.
Remark 1.5 ([4]). For z = 0, from inequality (1.1) we obtain for every ξ ∈ U,
g (ξ ) − g (0) 1 − g (0)g (ξ ) ≤ |ξ | and hence,
|ξ | + |g (0)|
|g (ξ )| ≤
1 + g (0)g (ξ )
.
Considering g (0) = a and ξ = z, then
|z | + |a|
|g (z )| ≤
1 + |a| |z |
,
for all z ∈ U. Theorem 1.6 ([5]). Let γ ∈ C, f ∈ S , f (z ) = z + a2 z 2 + · · ·. If
0 zf (z ) − f (z ) ≤ 1, zf (z )
∀z∈U
and 1
|γ | ≤ max
h
2
1 − |z |
· |z | ·
|z |≤1
|z |+|a2 | 1+|a2 ||z |
i,
then Fγ (z ) =
Z z
f (t )
γ dt
t
0
is in the class S . Theorem 1.7 ([5]). Let α, β, γ ∈ C, f ∈ S , f (z ) = z + a2 z 2 + · · ·. If
0 zf (z ) − f (z ) ≤ 1, zf (z )
∀ z ∈ U,
Re β ≥ Re α > 0 and 1
|γ | ≤ max
h
|z |≤1
1−|z |2Re α Re α
· |z | ·
|z |+|a2 | 1+|a2 ||z |
i,
then
Z z γ β1 β−1 f (t ) Gβ,γ (z ) = β t dt t
0
is in the class S . We consider the following two general integral operators: F[|δ|] (z ) =
Z z 0
f1 ( t ) t
α1
· ... ·
f[|δ|] (t ) t
α[|δ|]
dt ,
V. Pescar, N. Breaz / Applied Mathematics Letters 23 (2010) 1407–1411
1409
where δ ∈ C, |δ| 6∈ [0, 1), αi ∈ C, fi ∈ A, i = 1, [|δ|] and
α1 α γ1 Z z f[|γ |] (t ) [|γ |] γ −1 f1 (t ) · ... · t dt G[|γ |] (z ) = γ , t
0
t
γ ∈ C, |γ | 6∈ [0, 1), αi ∈ C, fi ∈ A, i = 1, [|γ |]. 2. Main results Theorem 2.1. Let δ ∈ C, |δ| ∈ 6 [0, 1), αi ∈ C, for i = 1, [|δ|], Re α1 · . . . · Re α[|δ|] = Re δ , 0 < Re δ ≤ 1. If fi ∈ A, i 2 fi (z ) = z + a2 z + · · · for i = 1, [|δ|] and
0 zfi (z ) − fi (z ) ≤ 1, ∀i = 1, [|δ|], z ∈ U, zfi (z ) |α1 | + · · · + α[|δ|] ≤ 1, α1 · . . . · α[|δ|] 1 α1 · . . . · α[|δ|] ≤ i, h |z |+|c | 1−|z |2Re δ | | max · z · Re δ 1+|c ||z |
(2.1)
(2.2)
(2.3)
|z |≤1
where
1 [|δ|] α1 a2 + · · · + α[|δ|] a2 |c | = , α1 · . . . · α[|δ|] then F[|δ|] (z ) =
Z z
f1 (t )
α1
t
0
· ... ·
f[|δ|] (t )
α[|δ|] dt
t
is in the class S . Proof. We have fi ∈ A, for all i = 1, [|δ|] and Let g be the function g (z ) =
f1 (z ) z
α1
fi (z ) 0, for all z α[|δ|] f[|δ|] (z )
· ... ·
6=
z
i = 1, [|δ|].
, z ∈ U. We have g (0) = 1.
Let us consider the function F 00 (z ) · [0|δ|] , α1 · . . . · α[|δ|] F[|δ|] (z )
h( z ) =
1
z ∈ U.
This function has the form 1
[|δ|] X
α1 · . . . · α[|δ|]
i=1
1
[|δ|] X
α1 · . . . · α[|δ|]
i=1
h( z ) =
αi
zfi0 (z ) − fi (z ) zfi (z )
.
Also, h(0) =
αi ai2 .
By using the relations (2.1) and (2.2) we obtain that |h(z )| < 1 and
1 [|δ|] α1 a2 + · · · + α[|δ|] a2 |h(0)| = = |c | . α1 · . . . · α[|δ|] Applying Remark 1.5 for the function h we obtain
F 00 (z ) |z | + |c | [|δ|] · ≤ , α1 · . . . · α[|δ|] F[0|δ|] (z ) 1 + |c | |z | 1
∀z ∈ U
1410
V. Pescar, N. Breaz / Applied Mathematics Letters 23 (2010) 1407–1411
and
1 − |z |2Re δ 1 − |z |2Re δ F[00|δ|] (z ) |z | + |c | ·z· 0 · |z | · , ≤ α1 · . . . · α[|δ|] Re δ F[|δ|] (z ) Re δ 1 + |c | |z |
∀z ∈ U.
(2.4)
Let us consider the function H : [0, 1] → R defined by 1 − x2Re δ
H (x) =
Re δ
·x·
x + |c | 1 + |c | x
x = |z | .
;
We have
H
1
> 0 ⇒ max H (x) > 0. x∈[0,1]
2
Using this result, from (2.4) we have
1 − |z |2Re δ F[00|δ|] (z ) |z | + |c | 1 − |z |2Re δ | | · z · · z · , ≤ α · . . . · α · max [|δ|] 1 Re δ |z |<1 F[0|δ|] (z ) Re δ 1 + |c | |z |
∀z ∈ U.
(2.5)
Applying the condition (2.3) in the form (2.5) we obtain that 1 − |z |2Re δ Re δ
F 00 (z ) [|δ|] · z · 0 ≤ 1, F[|δ|] (z )
∀z ∈ U,
and from Theorem 1.3 for β = 1, we obtain that F[|δ|] ∈ S .
Theorem 2.2. Let γ , δ ∈ C, |γ | 6∈ [0, 1), αi ∈ C, for i = 1, [|γ |], Re α1 · . . . · Re α[|γ |] = Re γ . If fi ∈ A, fi (z ) = z + ai2 z 2 + · · ·, for i = 1, [|γ |] and
0 zfi (z ) − fi (z ) ≤ 1, ∀i = 1, [|γ |], z ∈ U, zfi (z ) |α1 | + · · · + α[|γ |] ≤ 1, α1 · . . . · α[|γ |] Re γ ≥ Re δ > 0, 1 α1 · . . . · α[|γ |] ≤ i, h |z |+|c | 1−|z |2Re δ | | max · z · Re δ 1+|c ||z | |z |≤1
where
1 [|γ |] α1 a2 + · · · + α[|γ |] a2 |c | = , α1 · . . . · α[|γ |] then
Z z α1 α γ1 f[|γ |] (t ) [|γ |] γ −1 f1 (t ) G[|γ |] (z ) = γ t · ... · dt t
0
t
is in the class S . Proof. We consider the function h(z ) =
Z z 0
f1 (t )
α1
t
· ... ·
f[|γ |] (t )
α[|γ |]
t
Let us have the function h00 (z ) · , α1 · . . . · α[|γ |] h0 (z )
p(z ) =
1
z ∈ U.
dt .
(2.6)
(2.7)
(2.8)
V. Pescar, N. Breaz / Applied Mathematics Letters 23 (2010) 1407–1411
1411
The function p(z ) has the form 1
[|γ |] X
α1 · . . . · α[|γ |]
i =1
p(z ) =
αi
zfi0 (z ) − fi (z ) zfi (z )
.
By using the relations (2.6) and (2.7) we obtain |p(z )| < 1 and
1 [|γ |] α1 a2 + · · · + α[|γ |] a2 |p(0)| = = |c | . α1 · . . . · α[|γ |] Applying Remark 1.5 for the function h we obtain
00 h (z ) ≤ |z | + |c | , · α1 · . . . · α[|γ |] h0 (z ) 1 + |c | |z | 1
∀z ∈ U
and
2Re δ 1 − |z |2Re δ |z | + |c | h00 (z ) ≤ α1 · . . . · α[|γ |] 1 − |z | · z · · |z | · , Re δ 0 h (z ) Re δ 1 + |c | |z |
∀z ∈ U.
(2.9)
Let us consider the function Q : [0, 1] → R defined by Q (x) =
1 − x2Re δ Re δ
·x·
x + |c | 1 + |c | x
;
x = |z | .
We have Q 12 > 0 ⇒ maxx∈[0,1] Q (x) > 0. Using this result in (2.9), we have
1 − |z |2Re δ zh00 (z ) Re δ
2Re δ |z | + |c | ≤ α1 · . . . · α[|γ |] · max 1 − |z | | | , · z · h0 (z ) |z |<1 Re δ 1 + |c | |z |
∀z ∈ U.
(2.10)
Applying the condition (2.8) in the relation (2.10), we obtain that 1 − |z |2Re δ zh00 (z ) Re δ
h0 (z ) ≤ 1,
∀z ∈ U
and from Theorem 1.3, we obtain that G[|γ |] ∈ S .
References [1] Y.J. Kim, E.P. Merkes, On an integral of powers of a spirallike function, Kyungpook Math. J. 12 (2) (1972) 249–253. [2] J. Becker, Lownersche Differntialgleichung und quasikonform fortsetzbare schichte Functionen, J. Reine Angew. Math. 255 (1972) 23–43. [3] N.N. Pascu, An improvement of Becker’s univalence criterion, in: Proceedings of the Commemorative Session Simion Stoilow, University of Braşov, 1987, pp. 43–48. [4] G.M. Goluzin, Geometriceskaia teoria funktii kompleksnogo peremenogo, Moscova, 1966. [5] V. Pescar, On some integral operations which preserve the univalence, J. Math. XXX (1997) 1–10. (Punjab University).
Contents lists available at ScienceDirect
Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
On univalence of two integral operators Virgil Pescar a , Nicoleta Breaz b,∗ a
Department of Mathematics, ‘‘Transilvania’’ University of Braşov, Faculty of Mathematics and Computer Science, 500091 Braşov, Romania
b
Department of Mathematics, ‘‘1 Decembrie 1918’’ University of Alba Iulia, Faculty of Science, 510009 Alba Iulia, Romania
article
abstract
info
Article history: Received 26 April 2010 Accepted 21 July 2010
In this work we consider two integral operators. The integral operators were constructed on the basis of the fact that the number of functions from the composition of the operators is the entire part of a complex number modulus. © 2010 Elsevier Ltd. All rights reserved.
Keywords: Integral operator Univalence Integer part Modulus
1. Introduction Let us consider the open unit disk U and let A be the class of analytic functions defined by f (z ) = z + a2 z 2 + · · ·. We denote by S the class of univalent functions. Theorem 1.1 ([1]). If the function f belongs to the class S , then for any complex number γ , |γ | ≤ Fγ (z ) =
Z z
f (t ) t
0
1 , 4
the function
γ dt
is in the class S . Theorem 1.2 ([2]). If the function f is regular in the open unit disk U, f (z ) = z + a2 z 2 + · · · and
zf 00 (z ) ≤ 1, 1 − |z |2 0 f (z )
for all z ∈ U, then the function f is univalent in U. Theorem 1.3 ([3]). Let α be a complex number, Re α > 0 and f (z ) = z + a2 z 2 + · · · be a regular function in U. If 1 − |z |2Re α zf 00 (z )
f 0 (z ) ≤ 1,
Re α
for all z ∈ U, then for any complex number β , Re β ≥ Re α , the function z
Z
Fβ (z ) = β
t β−1 f 0 (t )dt
β1
0
is in the class S .
∗
Corresponding author. Fax: +40 258812630. E-mail addresses: [email protected] (V. Pescar), [email protected], [email protected] (N. Breaz).
0893-9659/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2010.07.007
1408
V. Pescar, N. Breaz / Applied Mathematics Letters 23 (2010) 1407–1411
Theorem 1.4 ([4]). If the function g is regular in U and |g (z )| < 1 in U, then for all ξ ∈ U, the following inequalities hold:
g (ξ ) − g (z ) ξ − z ≤ 1 − g (z )g (ξ ) 1 − z ξ
(1.1)
and
0 1 − |g (z )|2 g (z ) ≤ , 1 − |z |2 u and the equalities hold in the case g (z ) = 1z++uz , where || = 1 and |u| < 1.
Remark 1.5 ([4]). For z = 0, from inequality (1.1) we obtain for every ξ ∈ U,
g (ξ ) − g (0) 1 − g (0)g (ξ ) ≤ |ξ | and hence,
|ξ | + |g (0)|
|g (ξ )| ≤
1 + g (0)g (ξ )
.
Considering g (0) = a and ξ = z, then
|z | + |a|
|g (z )| ≤
1 + |a| |z |
,
for all z ∈ U. Theorem 1.6 ([5]). Let γ ∈ C, f ∈ S , f (z ) = z + a2 z 2 + · · ·. If
0 zf (z ) − f (z ) ≤ 1, zf (z )
∀z∈U
and 1
|γ | ≤ max
h
2
1 − |z |
· |z | ·
|z |≤1
|z |+|a2 | 1+|a2 ||z |
i,
then Fγ (z ) =
Z z
f (t )
γ dt
t
0
is in the class S . Theorem 1.7 ([5]). Let α, β, γ ∈ C, f ∈ S , f (z ) = z + a2 z 2 + · · ·. If
0 zf (z ) − f (z ) ≤ 1, zf (z )
∀ z ∈ U,
Re β ≥ Re α > 0 and 1
|γ | ≤ max
h
|z |≤1
1−|z |2Re α Re α
· |z | ·
|z |+|a2 | 1+|a2 ||z |
i,
then
Z z γ β1 β−1 f (t ) Gβ,γ (z ) = β t dt t
0
is in the class S . We consider the following two general integral operators: F[|δ|] (z ) =
Z z 0
f1 ( t ) t
α1
· ... ·
f[|δ|] (t ) t
α[|δ|]
dt ,
V. Pescar, N. Breaz / Applied Mathematics Letters 23 (2010) 1407–1411
1409
where δ ∈ C, |δ| 6∈ [0, 1), αi ∈ C, fi ∈ A, i = 1, [|δ|] and
α1 α γ1 Z z f[|γ |] (t ) [|γ |] γ −1 f1 (t ) · ... · t dt G[|γ |] (z ) = γ , t
0
t
γ ∈ C, |γ | 6∈ [0, 1), αi ∈ C, fi ∈ A, i = 1, [|γ |]. 2. Main results Theorem 2.1. Let δ ∈ C, |δ| ∈ 6 [0, 1), αi ∈ C, for i = 1, [|δ|], Re α1 · . . . · Re α[|δ|] = Re δ , 0 < Re δ ≤ 1. If fi ∈ A, i 2 fi (z ) = z + a2 z + · · · for i = 1, [|δ|] and
0 zfi (z ) − fi (z ) ≤ 1, ∀i = 1, [|δ|], z ∈ U, zfi (z ) |α1 | + · · · + α[|δ|] ≤ 1, α1 · . . . · α[|δ|] 1 α1 · . . . · α[|δ|] ≤ i, h |z |+|c | 1−|z |2Re δ | | max · z · Re δ 1+|c ||z |
(2.1)
(2.2)
(2.3)
|z |≤1
where
1 [|δ|] α1 a2 + · · · + α[|δ|] a2 |c | = , α1 · . . . · α[|δ|] then F[|δ|] (z ) =
Z z
f1 (t )
α1
t
0
· ... ·
f[|δ|] (t )
α[|δ|] dt
t
is in the class S . Proof. We have fi ∈ A, for all i = 1, [|δ|] and Let g be the function g (z ) =
f1 (z ) z
α1
fi (z ) 0, for all z α[|δ|] f[|δ|] (z )
· ... ·
6=
z
i = 1, [|δ|].
, z ∈ U. We have g (0) = 1.
Let us consider the function F 00 (z ) · [0|δ|] , α1 · . . . · α[|δ|] F[|δ|] (z )
h( z ) =
1
z ∈ U.
This function has the form 1
[|δ|] X
α1 · . . . · α[|δ|]
i=1
1
[|δ|] X
α1 · . . . · α[|δ|]
i=1
h( z ) =
αi
zfi0 (z ) − fi (z ) zfi (z )
.
Also, h(0) =
αi ai2 .
By using the relations (2.1) and (2.2) we obtain that |h(z )| < 1 and
1 [|δ|] α1 a2 + · · · + α[|δ|] a2 |h(0)| = = |c | . α1 · . . . · α[|δ|] Applying Remark 1.5 for the function h we obtain
F 00 (z ) |z | + |c | [|δ|] · ≤ , α1 · . . . · α[|δ|] F[0|δ|] (z ) 1 + |c | |z | 1
∀z ∈ U
1410
V. Pescar, N. Breaz / Applied Mathematics Letters 23 (2010) 1407–1411
and
1 − |z |2Re δ 1 − |z |2Re δ F[00|δ|] (z ) |z | + |c | ·z· 0 · |z | · , ≤ α1 · . . . · α[|δ|] Re δ F[|δ|] (z ) Re δ 1 + |c | |z |
∀z ∈ U.
(2.4)
Let us consider the function H : [0, 1] → R defined by 1 − x2Re δ
H (x) =
Re δ
·x·
x + |c | 1 + |c | x
x = |z | .
;
We have
H
1
> 0 ⇒ max H (x) > 0. x∈[0,1]
2
Using this result, from (2.4) we have
1 − |z |2Re δ F[00|δ|] (z ) |z | + |c | 1 − |z |2Re δ | | · z · · z · , ≤ α · . . . · α · max [|δ|] 1 Re δ |z |<1 F[0|δ|] (z ) Re δ 1 + |c | |z |
∀z ∈ U.
(2.5)
Applying the condition (2.3) in the form (2.5) we obtain that 1 − |z |2Re δ Re δ
F 00 (z ) [|δ|] · z · 0 ≤ 1, F[|δ|] (z )
∀z ∈ U,
and from Theorem 1.3 for β = 1, we obtain that F[|δ|] ∈ S .
Theorem 2.2. Let γ , δ ∈ C, |γ | 6∈ [0, 1), αi ∈ C, for i = 1, [|γ |], Re α1 · . . . · Re α[|γ |] = Re γ . If fi ∈ A, fi (z ) = z + ai2 z 2 + · · ·, for i = 1, [|γ |] and
0 zfi (z ) − fi (z ) ≤ 1, ∀i = 1, [|γ |], z ∈ U, zfi (z ) |α1 | + · · · + α[|γ |] ≤ 1, α1 · . . . · α[|γ |] Re γ ≥ Re δ > 0, 1 α1 · . . . · α[|γ |] ≤ i, h |z |+|c | 1−|z |2Re δ | | max · z · Re δ 1+|c ||z | |z |≤1
where
1 [|γ |] α1 a2 + · · · + α[|γ |] a2 |c | = , α1 · . . . · α[|γ |] then
Z z α1 α γ1 f[|γ |] (t ) [|γ |] γ −1 f1 (t ) G[|γ |] (z ) = γ t · ... · dt t
0
t
is in the class S . Proof. We consider the function h(z ) =
Z z 0
f1 (t )
α1
t
· ... ·
f[|γ |] (t )
α[|γ |]
t
Let us have the function h00 (z ) · , α1 · . . . · α[|γ |] h0 (z )
p(z ) =
1
z ∈ U.
dt .
(2.6)
(2.7)
(2.8)
V. Pescar, N. Breaz / Applied Mathematics Letters 23 (2010) 1407–1411
1411
The function p(z ) has the form 1
[|γ |] X
α1 · . . . · α[|γ |]
i =1
p(z ) =
αi
zfi0 (z ) − fi (z ) zfi (z )
.
By using the relations (2.6) and (2.7) we obtain |p(z )| < 1 and
1 [|γ |] α1 a2 + · · · + α[|γ |] a2 |p(0)| = = |c | . α1 · . . . · α[|γ |] Applying Remark 1.5 for the function h we obtain
00 h (z ) ≤ |z | + |c | , · α1 · . . . · α[|γ |] h0 (z ) 1 + |c | |z | 1
∀z ∈ U
and
2Re δ 1 − |z |2Re δ |z | + |c | h00 (z ) ≤ α1 · . . . · α[|γ |] 1 − |z | · z · · |z | · , Re δ 0 h (z ) Re δ 1 + |c | |z |
∀z ∈ U.
(2.9)
Let us consider the function Q : [0, 1] → R defined by Q (x) =
1 − x2Re δ Re δ
·x·
x + |c | 1 + |c | x
;
x = |z | .
We have Q 12 > 0 ⇒ maxx∈[0,1] Q (x) > 0. Using this result in (2.9), we have
1 − |z |2Re δ zh00 (z ) Re δ
2Re δ |z | + |c | ≤ α1 · . . . · α[|γ |] · max 1 − |z | | | , · z · h0 (z ) |z |<1 Re δ 1 + |c | |z |
∀z ∈ U.
(2.10)
Applying the condition (2.8) in the relation (2.10), we obtain that 1 − |z |2Re δ zh00 (z ) Re δ
h0 (z ) ≤ 1,
∀z ∈ U
and from Theorem 1.3, we obtain that G[|γ |] ∈ S .
References [1] Y.J. Kim, E.P. Merkes, On an integral of powers of a spirallike function, Kyungpook Math. J. 12 (2) (1972) 249–253. [2] J. Becker, Lownersche Differntialgleichung und quasikonform fortsetzbare schichte Functionen, J. Reine Angew. Math. 255 (1972) 23–43. [3] N.N. Pascu, An improvement of Becker’s univalence criterion, in: Proceedings of the Commemorative Session Simion Stoilow, University of Braşov, 1987, pp. 43–48. [4] G.M. Goluzin, Geometriceskaia teoria funktii kompleksnogo peremenogo, Moscova, 1966. [5] V. Pescar, On some integral operations which preserve the univalence, J. Math. XXX (1997) 1–10. (Punjab University).