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

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

( )

U

H ,A=

{

f ∈H

( ) ( )

U : f z = z+a₂z2 +...

}

the class of the analytical functions in the unit disk and the set

( )

( )

{

: isanalyticin , ≠0,∀ ∈ , 0 =1

}

= φ φ U φ z z U φ

D .

We consider the class of starlike functions of the order α on the unit disk, the univalent functions with the property

( )

( )

>α

′

z f

z f z

Re , respectively, the class of the convex functions on the order α in unit disk, the univalent functions with the property

( )

( )

+ >α

′′′ 1 Re

z f

z f z

. We denote these with S∗

( )

α

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+a₂z2 +... belongs to the class UCD

( )

α , α ≥0, if

( )

z zf

( )

z z U

f′ ≥ ′′ , ∈

Re α . (1)

If f ∈UCD

( )

α , then the following relation is true

( )

z zf

( )

z z U

f′ ≥α ′′ , ∈ (2)

( )

( )

α

1 ≤ ′′′z f

z f z

. (3)

In [3] we studied the class of univalent functions of the form

( )

∑

∞

= ≥

− =

2

0 ,

k

k k kz a a z z

f . (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

( )

[

]

∑

∞₌ + − ⋅ ≤

2

1 1

1

k

k a k

k α .

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

( )

2

TUCD

f ∈ .

d) A function of the form (4) is uniform convex if and only if

( )

2

TUCD

f ∈ .

e) A function f ∈TUCD

( )

α if and only if f can be written as

( )

∑

∞

( )

=

=

1

k k k f z z

f λ , (5)

where

∑

∞

= =

≥

1

1 ,

0

k k

k λ

λ , f₁

( )

z =z and

( )

(

)

[

]

, 2,3,... 1

1+ − =

−

= k

k k

z z

z f

k

k

α

Theorem 1.3. [2] Let α,β,γ,δ be real constants that satisfy the conditions

0 , 0 >

≥ β

α

(

α +δ

) (

= β +γ

)

>0. Let ρ₀ be so that 1 Re

Re

0 < <

≡

− γ_β ρ ρ

and we suppose that ρ∈

[ ]

ρ₀,1 exists, so that 0≤w

( )

ρ , where

( )

= 1 inf

{

H

( )

z : z <1

}

w

β

ρ and

( )

( )

( )−

−

= ₁ 1 z 2 1Re

z H

β ρ

Let ϕand φ∈Dsatisfy the conditions:

( )

( )

βρ γ

ϕϕ

δ + ′ ≥ +

z z z

Re (6)

and

( )

( )

β

( )

ρ

φφ z w z z

≤ ′

Re (7)

If

( )( )

_γ α ϕ δ β

φ γ

β 1 1

0

) ( ) ( )

( _

 

  +

= −

∫

f t t t dt z

z z f I

z

, then I

( )

S∗ ⊂S∗.

Theorem 1.4. [1] If 0≤α ≤1,α ≤β and if f ∈S∗, then the function

( )

( ) ...

1

0

1 = +

    

  

     

= −

∫

_{dt z}

t z f z

z F

z α _β

β ₍₈₎

is also an element of S*.

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

( )

( ) ...

1

0

1 = +

   



 + +

= + − +

∫

f t t dt z

z z

F

z _{α η}

η γ α

γ η

γ

α ₍₉₎

is starlike in U.

Theorem 1.6. [2] Let α,β,γ,δ and σ real number which satisfy σ

β

α < ≤

≤ ,0 ,0

0 and β +γ =α +δ >0 . Let ρ₀ defined in the theorem 1.3. and we suppose that exists ρ∈

[ ]

ρ₀,1 so that δ −σ ≥β +γ

2 and w

( )

ρ >0

( )

ρ β φφ z w

z z

≤ ′

) (

) (

Re . If f ∈S∗,g∈Kthen I

( )

f,g ∈S∗, where I

( )

f,g is defined

( )( )

α σ δ σ β

γ_φ γ

β 1

0

1

) ( ) ( ) (

, _

  

  +

=

∫

z _{f t g t t} − −_dt

z z z g f

I . (10)

Corollary 1.7. [1] Let 0≤α,0< β,0≤σ, γ >−β,β+γ =α +δ , and γ

σ α

δ + − ≥

2

2 . Letφ∈D which satisfies ( )

( )

0 )

(

Re w

z z z

β

φφ′ ≤ . If f,g∈Kthen

( )

_{f g ∈S}∗

I , , where I

( )

f,g is defined in (10). Main results

Theorem 2.1. Let ξ and δ complex number, ξ +δ >0 ,φ∈D. If f ∈Ais of the form (4) and

( )

( )

_φφ

( )

( )

δ ξ δ

ξ

+

+

+ Q

z z z z f

z zf

p '

'

, (11) while

( ) (

_{ξ δ}

) ( )

ξ δ− _φ

( )

ξ+δ

   

 

⋅ ⋅ +

=

∫

1

0

1 1

z

dt t t

t f f

F . (12)

Then F₁

( )

f ∈TUCD

( )

0 . Theorem 2.2. Let

( )( )

( )

ξ

( ) ( )

δ β

γ _φγ φ

β 1

0

1

2 

  



 _{⋅ ⋅}

⋅+

=

∫

z _{f t t t} −_dt

z z z f

F . (13)

If ξ ,β ,γ ,δ ∈R, ξ ≥0, β >0, β +γ =ξ +δ şi ρ₀ verifies the relation

1 Re

Re

0 ≤ <

≡

− _βγ ρ ρ (14)

and we suppose that exists ρ∈

[ ]

ρ₀,1 so that 0≤w

( )

ρ , where w verifies the relation

( )

( )

'

( )

Re

{

Re

( )

1

}

( )

( )( ) (

)

γ β

( )

γ β γ

β β γ

1

0

1

, 

  



 _{+ ⋅}

=

=_{I g z z}−

∫

z_{g t t} − _dt

z

G , g∈A (16)

and

( )

( )

( )

(

)

( )ρ β γ

γ β

β ρ

− +

− =

∫

+ − −

−

1 0

Re 1 2 1

Re 1 2

1 1

dt tz

t

z z

H , (17)

ϕ and φ∈D satisfy the relations

( )

( )

βρ γ

δ + ≥ +

z φ

z zφ'

Re (18)

and

( )

( )

β

( )

ρ

φφ z w

z z

⋅ ≤ '

Re (19)

then F₂

( )

f ∈TUCD

( )

0 .

Theorem 2.3. Let

( )

∑

∞

=

− =

2

k k kz a z z

f , a_k ≥0

a)If _f ∈_S* ⇒_F3

( )

_f ∈_TUCD

( )

1 _{, where}

( )( )

=

∫

z

( )

dt t

t f z f F

0

3 (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}

( )( )

=

∫

z

( )

dt t f z z f F

0 4

2

(21)

is the Libera operator. Proof.

c) f ∈S* ⇒F₅

( )

f ∈TUCD

( )

0 , where

( )( )

₌ +

∫

z _f

( )

_{t ⋅t} −_dt

z z f F

0

1 5

1 _γ

γγ (22)

1 − ≥

γ is the Bernardi operator. 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

( )( )

=

∫

z_

( )

_ dt t

t f z

f F

0 6

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

( )

* 6

* _{F f S}

S

f ∈ ⇒ ∈ .

According to the remark 1.2. a) we obtain F₆

( )

f ∈TUCD

( )

0 . Theorem 2.5. Let

( )( )

_β β

( )

β

1

0

1

7 

  



 _⋅

=

∫

z _{f t t}− _dt

z f

F , β>0. (24)

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

( )

( )

0

7 f TUCD

F ∈ .

Theorem 2.6. Let

( )( )

β

( )

β

γγ

β 1

0

1

8 

  

 

⋅ +

=

∫

z _{f t t}−_dt

z z

f

F , γ,β =1,2,3K. (25)

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

( )( )

( )

ξ

( )

δ ξ δ β

γγ

β 1

0

1

9 ,

    

  

⋅         +

=

∫

z _t + −_dt

t t g t

t f z

z g f

F . (26)

If _f ∈_S*_{, g}∈_{K, then}

( )

_,

( )

₀

9 f g TUCD

F ∈ .

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

( )( )

( )

ξ

( ) ( )

δ β

γ _φγ ϕ

β 1

0

1

10 

  



 _{⋅ ⋅}

⋅+

=

∫

z _{f t t t} −_dt

z z z f

F , f ∈A (27)

and the following conditions are satisfied:

δ γ β ξ ∈

φ , , , , , D

φ the complex numbers with the properties,

γ + β = δ + ξ >

β 0, , Re

(

ξ+δ

)

>0,

( )

( )

' 0

Re ≤

  

 _+γ

φφz z z

. (28) 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 ∈Ais the form (4) and

( )

( )

_φφ

( )

( )

δ ξ δ

ξ

+

+

+ Q

z z z z f

z zf

p '

'

, (29) and

( ) (

_{ξ δ}

) ( )

ξ δ− _φ

( )

ξ+δ

   



 _{+ ⋅ ⋅}

=

∫

1

0

1 11

z

dt t t

t f f

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

( )( )

( )

ξ

( ) ( )

δ β

γ _φγ φ

β 1

0

1

12 

  

 

⋅ ⋅ ⋅+

=

∫

z _{f t t t} −_dt

z z z f

F . (31)

If ξ ,β ,γ ,δ∈R, ξ ≥0, β >0, β +γ =ξ+δ and ρ₀ verifies the relation (14) and suppose that exists ρ∈

[ ]

ρ₀,1 so that 0≤w

( )

ρ , where w verifies the relation

( )

( )

'

( )

Re

{

Re

( )

1

}

Re ≥w ⋅ =Inf H z z <

z G

z zG

β ρ

β , (32)

where

( )

( )( ) (

)

γ β

( )

γ β

γ

β β γ

1

0

1

, 

  

 

⋅ +

=

=_{I g z z}−

∫

z_{g t t} − _dt

z

G , g∈A (33)

and

( )

( )

( )

(

)

( )ρ β γ

γ β

β ρ

− +

− =

∫

+ − −

−

1 0

Re 1 2 1

Re 1 2

1 1

dt tz

t

z z

H , (34)

ϕ şi φ∈D and satisfy the relations

( )

( )

βρ γ

δ + ≥ +

z φ

z zφ'

Re (35)

and

( )

( )

β

( )

ρ

φφ z w

z z

⋅ ≤ '

Re (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

( )

* ₁₂

( )

( )

0

12 f S F f TUCD

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:

( )

( )

βρ γ

δ

ξ _{+ + ≥ +}

z φ

z zφ' Re

2 , (37)

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

( )

( )

ξ

( ) ( )

σ δ σ β

γ_φ γ

β 1

0

1

13 , 

  



 + _{⋅ ⋅}

=

∫

z _{f t g t t} − −_dt

z z g f

F , (38)

δ γ β

ξ, , , and σ∈R, ξ ≥0, β >0, σ ≥0, β +γ =ξ +δ >0. Let ρ₀ from the theorem 2.10 and suppose that ρ∈

[ ]

ρ₀,1 so that

γ β σ

δ − ≥ +

2 , w

( )

ρ >0, (39) w given in the treorem 2.10.

Let φ∈D, so that

( )

( )

β w

( )

ρ z

φ

z zφ

⋅ ≤ '

Re . (40)

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, proved in [2] we obtain F₁₃

( )

f,g ∈S*. Applying the remark 1.2, a) we obtain

( )

,

( )

0

13 f g TUCD

F ∈ .

Theorem 2.13. Let

( )( )

( )

ξ

( ) ( )

σ δ σ β

γ_φ γ

β 1

0

1

14 , 

  



 + _{⋅ ⋅}

=

∫

z _{f t g t t} − −_dt

z z z g f

F , (41)

δ γ β

( )

0 ,

2

2− ≥ + ≥

+ξ σ βρ γ ρ

ρ w . (42)

Let φ∈D so that

( )

( )

β

( )

ρ

φφ z w z z

≤ '

Re . (43)

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

( )

_{f g ∈S}∗

F₁₄ , . Applying the remark 1.2, a) we obatin F₁₄

( )

f,g ∈TUCD

( )

0 .

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.

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