Aticol revrez. 90KB Jun 04 2011 12:08:22 AM

ACTA UNIVERSITATIS APULENSIS

47

ON UNIVALENT INTEGRAL OPERATOR

by

Daniel Breaz and Nicoleta Breaz

Abstract.Let S be the class of regular and univalent function f

( )

z =z+a₂z2 +..., in the unit disc, U =

{

z: z <1

}

.We prove new univalence criteria for the integral operator F_αβ.

Theorem 1. If the function f is regular in unit disc U, f

( )

z = z+a₂z2 +...,and

( )

( )

( )

( )

z U z

f z f z

z ≤ ∀ ∈

′′′ ⋅

− 1,

1 2 , (1)

then the function f is univalent in U.

Theorem 2. If the function g is regular in U and g

( )

z <1 in U, then for all ξ∈Uand U

z∈ the following inequalities hold

( ) ( )

( ) ( )

ξ ξ ξ ξ

z z

g z g

z g g

− − ≤ −

−

1

1 , (2) and

( )

( )

2 2

1 1

z z g z

g

− − ≤

′ , (3)

the equalities hold only in case

( )

z u

u z z g

+ + =

1

ε where

ε

=1 and u <1.

Remark A For z=0 , from inequality (2) we obtain for every ξ∈U

( ) ( )

( ) ( )

ξ ξ

ξ _≤

− −

g g

g g

0 1

0

, (4) and, hence

( )

ξ ξ

( )

( )

_ξ 0 1

0 g

g g

+ +

≤ , (5)

Considering g

( )

0 =a and ξ =z then

( )

z a

a z z g

+ + ≤

1 , (6)

ACTA UNIVERSITATIS APULENSIS

48

Theorem 3. Let γ be a complex number and the function h∈S,h

( )

z =z+a₂z2 +...,. If

( ) ( )

( )

( )

z U

z zh

z h z h z

∈ ∀ ≤ −

′ _{1, , (7)}

for all z∈U and the constant γ satisfies the condition

( )

_

  

  

⋅ +

+ ⋅ ⋅ − ≤

≤ c z

c z z z

z 1

1 1

2 1

max

γ , (8)

then the function

( )

( )

dt S

t t f z F

z

∈ ⋅       =

∫

0

γ

γ , (10)

Theorem 4. Let α,β∈C, f,g∈S ,f

( )

z =z+a₂z2 +...,g

( )

z =z+b₂z2+...,. If

( ) ( )

( )

( )

z U

z zf

z f z f z

∈ ∀ ≤ − ′

,

1 , (11)

( ) ( )

( )

( )

z U

z zg

z g z g z

∈ ∀ ≤ − ′

,

1 , (12)

1 1 1

< + _β

α , (13)

( )

_

  

  

⋅ +

+ ⋅ ⋅ − ≤

⋅

≤ c z

c z z z

z 1

1 1

2 1

max

β

α , (14)

where

β

α β

α ⋅ + = a2 b2

c , (15)

then

( )

( )

( )

dt S

t t g t

t f z F

z

∈      ⋅      

=

∫

α β

αβ

0

Proof:

, ,g S

f ∈ and

( )

≠0,

( )

≠0 z

z g z

z f

.

For z=0 we are

( )

( )

 =1      ⋅     

 α β

z z g z

z f

ACTA UNIVERSITATIS APULENSIS

49

We consider the function

( )

( )

( )

z F z F z h αβ αβ β α ′ ′′ ⋅ ⋅

= 1 , where α⋅β satisfy (14).

We calculate the derivative by order 1 and 2 for F_αβ.

We are:

( )

( )

( )

β α αβ       ⋅       = ′ z z g z z f z F

( )

( )

α

( ) ( ) ( )

β

( )

β

( ) ( ) ( )

α αβ α β       ⋅ − ⋅       +       ⋅ − ′ ⋅       = ′′ − − z z f z z g z zg z z g z z g z z f z zf z z f z F ₂ 1 2 1

Then h(z) are the form:

( )

( )

( )

( )

( ) ( ) ( )

( )

( )

_ +      ⋅             ⋅ − ′ ⋅       ⋅ ⋅ = ′ ′′ ⋅ ⋅ = − β α β α αβ αβ α β α β α z z g z z f z z g z z f z zf z z f z F z F z h 2 1 1 1

( )

( ) ( ) ( )

( )

α

( )

β α β β β α       ⋅             ⋅ − ⋅       ⋅ ⋅ + − z z g z z f z z f z z g z zg z z g 2 1 1 .

( ) ( )

( )

( ) ( )

zg

( )

z

z g z zg z zf z f z f

z ′ −

⋅ ⋅ ⋅ + − ′ ⋅ ⋅ ⋅

= _α1_β α _α1_β β .

We are h

( )

0 1 a₂ 1 βb₂ αβ α

β

α⋅ ⋅ + ⋅

= and the condition (11) and (12)

But

( )

( ) ( )

( )

− + ⋅ ⋅ ⋅ ′

( ) ( )

( )

− ≤ ′ ⋅ ⋅ ⋅ = z zg z g z zg z zf z f z f z z h β β α α β α 1 1

( ) ( )

( )

⋅ ′

( ) ( )

( )

− ≤ 1 + 1 <1 ⋅ + − ′ ⋅ ⋅ ≤ _αα_{β α}β_{β α β} z zg z g z zg z zf z f z f z

from (13) and h

( )

z <1.

( )

a b c

h = + =

αββ

α 2 2

0

Applied Remark A for the function h obtained:

( )

( )

z U z

c c z z

h ∀ ∈

⋅ +

+

≤ ,

1

But

( )

( )

ACTA UNIVERSITATIS APULENSIS

50

And we have

( )

( )

+ ⋅

( )

∀ ∈ ⇔

+ ≤ ′

′′ ⋅

⋅ c z z U

c z

z F

z F

, 1

1

αβ αβ

β α

( )

( )

+ ⋅

( )

∀ ∈ ⇔

+ ⋅ ⋅ ≤ ′

′′

⇔ z U

z c

c z

z F

z F

, 1

β α

αβ αβ

( )

( )

( )

( )

( )

z U

z c

c z z z z

F z F z

z ∀ ∈

⋅ +

+ ⋅ ⋅ − ⋅ ⋅ ≤ ′

′′ ⋅ ⋅ −

⇔ ,

1 1

1 2 α β 2

αβ

αβ _{.(applied th.1) (16)}

Let’s consider the function H:

[ ]

0,1 →R,

( )

( )

x z x

c c x x x x

H =

⋅ +

+ ⋅ ⋅ −

= ,

1

1 2

[ ]

( )

0 0

2 1 8 3 2 1 1

2 1 2 1 4 1 1 2 1

max

1 , 0

> ⇒

> + + ⋅ = ⋅ +

+ ⋅ ⋅       − =      

∈ H x

c c

c c H

x

.

Using this result in (16) we have:

( )

( )

( )

[ ]

( )

c z

( )

z U

c z z z z

F z F z z

x  ∀ ∈

 

  

⋅ +

+ ⋅ ⋅ − ⋅

⋅ ≤ ′

′′ ⋅ ⋅ −

∈ 1 1 ,

1 2

1 , 0 2

max

β α

αβ αβ

and (14) implies

( )

( )

( )

( )

z U z

F z F z

z ≤ ∀ ∈

′ ′′ ⋅

− 1,

1 2

αβ

αβ _{and using the theorem 1 obtained F∈S.}

Remark B. For g

( )

z =z,β∈C,β >1, we obtained theorem 3.

REFERENCES

[1] V.Pescar- On some integral operations which preserve the univalence,,Journal of Mathematics, Vol. xxx (1997) pp.1-10, Punjab University

[2] J. Becker, Lownersche Differentialgleichung und quasikonform fortsezbare schichte Funktionen, J. Reine Angew. Math. 225 (1972), 23-43.

[3] N.N. Pascu- An improvement of Becker’s univalence criterion, Proceedings of the Commemorative Session Simion Stoilow, Braşov, (1987), 43-48.

[4] N.N. Pascu, V. Pescar, On the integral operators of Kim-Merkens and

Pfaltzgraff, Studia(Mathematica), Univ. Babeş-Bolyai, Cluj-Napoca, 32, 2(1990), 185-192.

Authors: