Breaz Daniel. 147KB Jun 04 2011 12:08:27 AM

Proceedings of the International Conference on Theory and Applications of
Mathematics and Informatics - ICTAMI 2004, Thessaloniki, Greece

INTEGRAL OPERATORS ON THE TUCD( α )-CLASS
by
Daniel Breaz
Abstract. In this paper we present a few univalency conditions for various integral operators
on class TUCD (α ) . At first, we make a brief presentation of class TUCD (α ) and of some of
its properties, as well as a number of matters connected to some integral operators studied on
this class.

Introduction.
Let U = {z : z < 1} be the unit disk, respectively the class of olomorphic
functions
on
the
unit
disk,
denoted
by
2

H (U ) , A = f ∈ H (U ) : f ( z ) = z + a 2 z + ... the class of the analytical
functions
in
the
unit
disk
and
the
set
D = {φ : φ is analytic in U , φ ( z ) ≠ 0, ∀z ∈ U , φ (0 ) = 1}.
We consider the class of starlike functions of the order α on the unit
zf ′(z )
disk, the univalent functions with the property Re
> α , respectively, the
f (z )
class of the convex functions on the order α in unit disk, the univalent
zf ′′( z )
+ 1 > α . We denote these with S ∗ (α )
functions with the property Re
f ′( z )

and K (α ) , and for α = 0 we obtain the class of the starlike funcions S ∗ ,
respectively the class of convex functions K .
We say that the function f with the form f ( z ) = z + a 2 z 2 + ... belongs
to the class UCD(α ) , α ≥ 0 , if
Re f ′( z ) ≥ α zf ′′( z ) , z ∈ U .
(1)

{

}

If f ∈ UCD(α ) , then the following relation is true
f ′( z ) ≥ α zf ′′( z ) , z ∈ U
which can also be written as

40

(2)

Daniel Breaz - Integral operators on the TUCD( α )-class

zf ′′( z ) 1
≤ .
(3)
f ′( z ) α
In [3] we studied the class of univalent functions of the form
f (z ) = z − ∑ ak z k , ak ≥ 0 .
∞

(4)

We denote TUCD (α ) , the functions from UCD(α ) of the form (4). Next we
consider all the functions from this paper of the form (4).
Theorem 1.1. [3] A function of the form (1) belongs to the class TUCD (α ) if
and only if
k =2

∑ k [1 + α (k − 1)]⋅ a
∞

k =2

k

≤ 1.

Remark 1.2. [3]
a) A function of the form (4) is starlike if and only if f ∈ TUCD (0) .
b) A function of the form (4) is convex if and only if f ∈ TUCD (1) .
c) A function of the form (4) is convex by the order 1/2 if and only if
f ∈ TUCD (2) .
d) A function of the form (4) is uniform convex if and only if
f ∈ TUCD (2) .
e) A function f ∈ TUCD (α ) if and only if f can be written as
f (z ) = ∑ λ k f k (z ) ,
∞

k =1

where λ k ≥ 0, ∑ λ k = 1 , f 1 ( z ) = z and f k ( z ) = z −
∞

k =1

(5)
k [1 + α (k − 1)]
zk

, k = 2,3,...

Theorem 1.3. [2] Let α , β , γ , δ be real constants that satisfy the conditions
Re γ
α ≥ 0, β > 0 (α + δ ) = (β + γ ) > 0 . Let ρ 0 be so that −
≡ ρ0 < ρ < 1
Re β
and we suppose that ρ ∈ [ρ 0 ,1] exists, so that 0 ≤ w( ρ ) , where

1
inf {H ( z ) : z < 1} and H ( z ) =
w( ρ ) =
Re β

∫t
1

0

41

(1 − z )2( ρ −1) Re β

β +γ −1

(1 + tz )

2 ( ρ −1) Re β

.
dt

Daniel Breaz - Integral operators on the TUCD( α )-class

Let ϕ and φ ∈ D satisfy the conditions:

δ + Re
and
Re

zϕ ′(z )
≥ βρ + γ
ϕ (z )

(6)

zφ ′( z )
≤ βw( ρ )
φ (z )

(7)

( )

β
 β +γ z α
δ −1 
If I ( f )( z ) =  γ
f
t
t
t
dt
, then I S ∗ ⊂ S ∗ .
(
)
(
)
ϕ

∫
 z φ ( z) 0

1

Theorem 1.4. [1] If 0 ≤ α ≤ 1, α ≤ β and if f ∈ S ∗ , then the function
 β −1 z  f ( z )  α  β
F (z ) =  z ∫ 
 dt  = z + ...
t  

0
1

(8)

is also an element of S*.

Theorem 1.5. [1] If α > 0,η ≥ 0, γ + η ≥ 0 and if f ∈ S ∗ , then the function

α + γ + η

F (z ) = 
f α (t )t γ +η −1 dt 

γ
∫
0
 z

z

α +η
1

= z + ...

(9)

is starlike in U.
Theorem 1.6. [2]
Let α , β , γ , δ and σ real number which satisfy
0 ≤ α ,0 < β ,0 ≤ σ and β + γ = α + δ > 0 . Let ρ 0 defined in the theorem 1.3.
and we suppose that exists ρ ∈ [ρ 0 ,1] so that δ −

where w( ρ ) Re β ≡ Inf {Re H ( z ) : z < 1}.

42

Let

σ

≥ β + γ and w( ρ ) > 0
2
φ ∈D
which
satisfies

Daniel Breaz - Integral operators on the TUCD( α )-class
zφ ′( z )
≤ βw( ρ ) . If
φ ( z)
defined
Re

f ∈ S ∗ , g ∈ K then I ( f , g ) ∈ S ∗ , where I ( f , g ) is

β
 β +γ z α
σ
δ −σ −1 
(
)
(
)
I ( f , g )( z ) =  γ
f
t
g
t
t
dt
 . (10)
∫
 z φ ( z) 0

Corollary 1.7. [1] Let 0 ≤ α ,0 < β ,0 ≤ σ , γ > − β , β + γ = α + δ , and
α σ
zφ ′( z )
δ + − ≥ γ . Let φ ∈ D which satisfies Re
≤ βw(0) . If f , g ∈ K then
2 2
φ ( z)
I ( f , g ) ∈ S ∗ , where I ( f , g ) is defined in (10).
1

Main results

Theorem 2.1. Let ξ and δ complex number, ξ + δ > 0, φ ∈ D .
If f ∈ A is of the form (4) and
ξzf ' ( z ) zφ ' ( z )
+
+ δ p Qξ +δ ,
(11)
f (z )
φ (z )
while

z

 ξ +δ
F1 ( f ) = (ξ + δ )∫ f ξ (t ) ⋅ t δ −1 ⋅ φ (t )dt  .
0


Then F1 ( f ) ∈ TUCD (0) .
Theorem 2.2. Let
1

(12)

β
 β +γ z ξ
δ −1 
(
)
φ
(
)
F2 ( f )( z ) =  γ
f
t
t
t
dt
⋅
⋅
 . (13)
∫
φ
(
)
z
z
⋅
0


If ξ , β , γ , δ ∈ R , ξ ≥ 0 , β > 0 , β + γ = ξ + δ şi ρ 0 verifies the relation
Re γ
−
≡ ρ0 ≤ ρ < 1
(14)
Re β
and we suppose that exists ρ ∈ [ρ 0 ,1] so that 0 ≤ w( ρ ) , where w verifies the
relation
zG ' ( z )
Re β
≥ w( ρ ) ⋅ Re β = Inf {Re H ( z ) z < 1},
(15)
G (z )
where
1

43

Daniel Breaz - Integral operators on the TUCD( α )-class
z


G ( z ) = I β ,γ ( g )( z ) = (β + γ )z −γ ∫ g β (t ) ⋅ t γ −1 dt 
0



1

2 ( ρ −1) Re β
1 − z)
(
H (z ) = 1
−γ
2 ( ρ −1) Re β
β +γ −1
(
)
t
1
tz
dt
+
∫0

and

β

, g∈A

,

(16)

(17)

ϕ and φ ∈ D satisfy the relations
δ + Re

and
Re

then F2 ( f ) ∈ TUCD (0 ) .

zφ' ( z )
≥ βρ + γ
φ( z )

zφ ' ( z )
≤ β ⋅ w( ρ )
φ (z )

(18)

(19)

Theorem 2.3. Let f ( z ) = z − ∑ a k z k , a k ≥ 0
∞

a)If f ∈ S ⇒ F3 ( f ) ∈ TUCD (1) , where
k =2

*

F3 ( f )( z ) = ∫
z

0

f (t )
dt
t

(20)

is the Alexander operator.
Proof.
We know that if f ∈ S * ⇒ F3 ( f ) ∈ K . Considering the remark 1.2. b), we
obtain F3 ( f ) ∈ K ⇔ F3 ( f ) ∈ TUCD (1) .

b) f ∈ S * ⇒ F4 ( f ) ∈ TUCD (0 ) , where
2 z
F4 ( f )( z ) = ∫ f (t )dt
z 0
is the Libera operator.

(21)

Proof.
We know that if f ∈ S * ⇒ F4 ( f ) ∈ S ∗ . Considering the remark 1.2. a), we
obtain F4 ( f ) ∈ S ∗ ⇔ F4 ( f ) ∈ TUCD (0) .

44

Daniel Breaz - Integral operators on the TUCD( α )-class
c) f ∈ S * ⇒ F5 ( f ) ∈ TUCD (0 ) , where

1+ γ
zγ
γ ≥ −1 is the Bernardi operator.
F5 ( f )( z ) =

∫ f (t ) ⋅ t
z

γ −1

dt

(22)

0

Proof.
We know that if f ∈ S * ⇒ F5 ( f ) ∈ S ∗ . Considering the remark 1.2. a), we

obtain F5 ( f ) ∈ S ∗ ⇔ F5 ( f ) ∈ TUCD (0 ) .
Theorem 2.4. Let

 f (t ) 
dt
F6 ( f )( z ) = ∫ 
t 
0 
z

β

(23)

and 0 ≤ β ≤ 1 .
If f ∈ S * then F6 ( f ) ∈ TUCD (0) .
Proof.
Considering the theorem 1.4, for β = 1, α ≡ β we obtain f ∈ S * ⇒ F6 ( f ) ∈ S * .
According to the remark 1.2. a) we obtain F6 ( f ) ∈ TUCD (0) .
Theorem 2.5. Let

 z
 β
(24)
F7 ( f )( z ) =  β ∫ f β (t ) ⋅ t −1 dt  , β > 0 .
 0

*
If f ∈ S then F7 ( f ) ∈ TUCD (0) .
Proof.
Applying the theorem 1.5 for this integral operator for γ = 0,η = 0, β = α we
obtain f ∈ S * ⇒ F7 ( f ) ∈ S * . But the remark 1.2, a) we obtain
F7 ( f ) ∈ TUCD (0) .
1

Theorem 2.6. Let


β + γ z β
F8 ( f )( z ) =  γ ∫ f (t ) ⋅ t −1 dt 

 z 0
*
If f ∈ S then F8 ( f ) ∈ TUCD(0) .

45

1

β

, γ , β = 1,2,3K . (25)

Daniel Breaz - Integral operators on the TUCD( α )-class
Proof.
We have f ∈ S * ⇒ F8 ( f ) ∈ S * , according to the theorem 1.5. Applying the
remark 1.2, a), we obtain F8 ( f ) ∈ TUCD(0) .
Theorem 2.7. Let
 β + γ z  f (t )  ξ  g (t )  δ ξ +δ −1 
dt 
F9 ( f , g )( z ) =  γ ∫ 
 ⋅t
 

 z 0  t   t 

1

β

. (26)

If f ∈ S * , g ∈ K , then F9 ( f , g ) ∈ TUCD(0) .
Proof.
In [2] we proved that if f ∈ S * , g ∈ K ⇒ F9 ( f , g ) ∈ S ∗ . Applying the remark
1.2 a) we obtain F9 ( f , g ) ∈ TUCD(0) .
Theorem 2.8. Let
β
 β +γ z ξ
δ −1 
(
)
ϕ
(
)
⋅
⋅
F10 ( f )( z ) =  γ
f
t
t
t
dt
 , f ∈A
∫
 z ⋅ φ (z ) 0

and the following conditions are satisfied:
φ, φ ∈ D, ξ, β, γ, δ the complex numbers with the properties,
1

(27)

 zφ' ( z ) 
β > 0, ξ + δ = β + γ , Re(ξ + δ ) > 0 , Re 
+ γ ≤ 0 .
(28)
 φ( z )

Then F10 ( f ) ∈ TUCD (0) .
Proof.
If f ∈ A satisfies the conditions of this theorem then F10 ( f ) ∈ S * , the result
proved in [2].
Applying the remark 1.2, a) we obtain F10 ( f ) ∈ TUCD (0) .

Theorem 2.9. Let ξ şi δ complex number, ξ + δ > 0, φ ∈ D .
If f ∈ A is the form (4) and
ξzf ' ( z ) zφ ' ( z )
+
+ δ p Qξ +δ ,
(29)
φ (z )
f (z )
and
z

 ξ +δ
F11 ( f ) = (ξ + δ )∫ f ξ (t ) ⋅ t δ −1 ⋅ φ (t )dt  .
0


1

46

(30)

Daniel Breaz - Integral operators on the TUCD( α )-class
Then F11 ( f ) ∈ TUCD (0) .
Proof.
If f ∈ A are the form (4) and verifies the conditions of the theorem 2.1, we
obtain F11 ( f ) ∈ S * ⇔ F11 ( f ) ∈ TUCD (0) , also considering the remark 1.2, a).
Theorem 2.10. Let

β
 β +γ z ξ
δ −1 
(
)
φ
(
)
F12 ( f )( z ) =  γ
f
t
t
t
dt
(31)
⋅
⋅
 .
∫
 z ⋅ φ (z ) 0

If ξ , β , γ , δ ∈ R , ξ ≥ 0 , β > 0 , β + γ = ξ + δ and ρ 0 verifies the relation
(14) and suppose that exists ρ ∈ [ρ 0 ,1] so that 0 ≤ w( ρ ) , where w verifies the
relation
zG ' ( z )
Re β
≥ w( ρ ) ⋅ Re β = Inf {Re H ( z ) z < 1},
(32)
G (z )
where
1

z


G ( z ) = I β ,γ ( g )( z ) = (β + γ )z −γ ∫ g β (t ) ⋅ t γ −1 dt 
0



and

H (z ) =

1

(1 − z )2( ρ −1) Re β
−γ
1
2 ( ρ −1) Re β
β +γ −1
dt
∫0 t (1 + tz )

β

,

, g∈A

(33)

(34)

ϕ şi φ ∈ D and satisfy the relations
δ + Re

and
Re

zφ' ( z )
≥ βρ + γ
φ( z )

zφ ' ( z )
≤ β ⋅ w( ρ )
φ (z )

(35)

(36)

then F12 ( f ) ∈ TUCD (0) .
Proof.
Let f ∈ A and satisfies the above conditions according to theorem 1.3 and the
remark 1.2, a) we obtain F12 ( f ) ∈ S * ⇔ F12 ( f ) ∈ TUCD (0) .

47

Daniel Breaz - Integral operators on the TUCD( α )-class
Theorem 2.11. Let the integral operator F12 ( f ) defined in the theorem 2.10,
which verifies the conditions of the theorem 2.10, and the condition (35) from
theorem 2.10 becomes the folloving:
zφ' ( z )
ξ
+ δ + Re
≥ βρ + γ ,
(37)
φ( z )
2
then if f ∈ K ⇒ F12 ( f ) ∈ TUCD(0) .
Proof.
If f ∈ K , has the form (4) and verifies the conditions of this theorem we
obtain F12 ( f ) ∈ S * . Applying the remark 1.2, a) we obtain F12 ( f ) ∈ TUCD (0) .
Theorem 2.12. Let
β
 β +γ z ξ
σ
δ −σ −1 
F13 ( f , g ) =  γ
∫ f (t ) ⋅ g (t ) ⋅ t dt  , (38)
 z φ (z ) 0
ξ , β , γ , δ and σ ∈ R , ξ ≥ 0 , β > 0 , σ ≥ 0 , β + γ = ξ + δ > 0 . Let ρ 0 from
the theorem 2.10 and suppose that ρ ∈ [ρ 0 ,1] so that
1

σ

≥ β + γ , w( ρ ) > 0 ,
(39)
2
w given in the treorem 2.10.
Let φ ∈ D , so that
zφ' ( z )
Re
(40)
≤ β ⋅ w( ρ ) .
φ( z )
If f ∈ S * , g ∈ K , then F13 ( f , g ) ∈ TUCD (0) .
Proof.
We consider f ∈ S * , g ∈ K of the form (4). According to the theorem 1.7,

δ−

proved in [2] we obtain F13 ( f , g ) ∈ S * . Applying the remark 1.2, a) we obtain
F13 ( f , g ) ∈ TUCD (0) .
Theorem 2.13. Let

β
 β +γ z ξ
σ
δ −σ −1 
⋅
⋅
F14 ( f , g )( z ) =  γ
f
t
g
t
t
dt
(41)
(
)
(
)
 ,
∫
φ
(
)
z
z
0


ξ , β , γ , δ and σ ∈ R , α ≥ 0 , β > 0 , σ ≥ 0 , β + γ = ξ + δ > 0 , ρ 0 , w( ρ ) as in
the theorem 2.10 and exists ρ ∈ [ρ 0 ,1] so that
1

48

Daniel Breaz - Integral operators on the TUCD( α )-class

Let φ ∈ D so that

ρ+

ξ
2

−

σ
2

≥ βρ + γ , w( ρ ) ≥ 0 .

(42)

zφ ' ( z )
(43)
≤ βw( ρ ) .
φ (z )
If f , g ∈ K of the form (4) then F14 ( f , g ) ∈ TUCD (0) .
Proof.
If f , g ∈ K according to the corrolary 1.6, the result proved in [1] we obatin
F14 ( f , g ) ∈ S ∗ . Applying the remark 1.2, a) we obatin F14 ( f , g ) ∈ TUCD (0) .
Re

References

1.Miller S.S., Mocanu P.T.- Differential Subordinations. Theory and
Applications, Marcel Dekker, Inc., New York.Basel, 2000.
2.Miller S.S., Mocanu P.T.- Classes of univalent integral operators, Journal of
Math. Analysis and Applications, vol. 157, No. 1, May 1, 1991, Printed in
Belgium, 147-165.
3.Rosy T., Stephen A.B., Subramanian K.G., Silverman H.- Classes of convex
functions, Internat. J. Math.& Math. Sci., vol 23, No. 12(2000), 819-825.
4.Silverman H.- Univalent functions with negative coefficients, Proc. Amer.
Math. Soc. 51(1975), 109-116.

Author:

Daniel Breaz – “1 Decembrie 1918” University of Alba Iulia, E-mail address:
dbreaz@uab.ro

49