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A PRESERVING PROPERTY OF THE ALEXANDER OPERATOR

by

Mugur Acu and Dorin Blezu

Abstract.The H

( )

U be the set of functions which are regular in the unit disc U and

{

∈ ( ), (0)=0, '(0)=1

}

= f H U f f

A .

Let denote by I the Alexander operator I:A→H(U) as:

) 1 ( ) ( ) ( ) (

0

∫

= =

z

dt t

t F z IF z

f .

The purpose of this note is to show that the n-uniform starlike functions of order γ and type α and the n-uniform close to convex functions of order γ and type α are preserved by the Alexander operator.

1. Preliminary results

Definition 2.1. Let

D

n be the Sălăgean differential operator defined as:

N n A A

Dn: → , ∈ and

(

( )

)

. )

(

) ( ) ( ) (

) ( ) (

1 ' 1

0

z f D D z f D

z zf z Df z f D

z f z f D

n

n = −

= =

=

Definition 2.2. [2] Let f ∈A, we say that f is n-uniform starlike function of order

γ

and type

α

if

U z z

f D

z f D z

f D

z f D

n n

n n

∈ +

− ⋅

≥   

 + +

, 1 ) (

) ( )

( ) ( Re

1 1

γ α

where α ≥0, γ∈[−1,1), α+γ ≥0, n∈N. We denote this class with US_n(α,γ). Remark 2.1. Geometric interpretation: f ∈USn(α,γ) if and only if

) (

) ( 1

z f D

z f D

n n+

take all values in the convex domain included in right half plane ∆_α,γ , where ∆_α,γ is a eliptic region for α >1, a parabolic region for α =1, a hyperbolic region for

1

2

Definition 2.3. [1] Let f∈A, we say that f is n-uniform close to convex function of order

γ

and type

α

in respect to the functions n-uniform starlike of order

γ

and type )α g(z , where α ≥0, γ ∈[−1,1), α+γ ≥0, if

U z z

g D

z f D z

g D

z f D

n n

n n

∈ +

− ⋅

≥   

 + +

, 1 ) (

) ( )

( ) ( Re

1 1

γ α

where α ≥0, γ∈[−1, 1), α+γ ≥0, n∈N. We denote this class with )

, (α γ

n

UCC .

Remark 2.2. Geometric interpretation: f ∈UCCn(α,γ) if and only if

) (

) ( 1

z g D

z f D

n n+

take all values in the convex domain included in right half plane ∆_α,γ , where ∆_α,γ is a eliptic region for α >1, a parabolic region for α =1, a hiperbolic region for 0<α <1, the half plane u>γ for α =0.

The next two theorems are results of the so called “admissible functions method” introduced by P. T. Mocanu and S. S. Miller (see [3], [4], [5]).

Theorem 2.1. Let h convex in U and Re[βh(z)+γ]>0, z∈U. If p∈H(U) with )p(0)=h(0 and p satisfied the Briot-Bouquet differential subordination

) ( )

( ) ( )

(

'

z h z

p z zp z

p p

γ

β +

+ , then p(z)ph(z).

Theorem 2.2. Let q convex in U and j:U →C with Re[j(z)]>0. If p∈H(U) and p satisfied p(z)+ j(z)⋅zp'(z)pq(z), then p(z)pq(z).

2. Main results

Theorem 3.1. If F(z)∈USn(α,γ), with α ≥0 and γ >0, then )

, ( ) ( )

(z IF z USn α γ

f = ∈ with α ≥0 and γ >0.

Proof. We know that F(z)∈USn(α, γ) if and only if

) (

) ( 1

z F D

z F D

n n+

3

By differentiating = =

∫

z dt t t F z IF z f 0 ) ( ) ( )

( we obtain: F(z)=zf'(z). By means of the application of the linear operator Dn+1 we obtain:

( )

( ) )

( 1 '

1 z zf D z F

Dn+ = n+ or Dn+1F(z)=Dn+2f(z).

Similarly, by means of the application of the linear operator Dn we obtain: )

( )

(z D 1f z F

Dn = n+ . Thus ) 2 ( ) ( ) ( ) ( ) ( 1 2 1 z f D z f D z F D z F D n n n n + + + = .

With notation ( )

) ( ) ( 1 z p z f D z f D n n = +

, where p(z)=1+p₁z+... we have

(

)

(

)

2

2 1 2 ' ) ( ) ( ) ( ) ( ) ( z f D z f D z f D z f D z zp n n n

n+ ⋅ − +

= . ) ( ) ( ) ( ) ( ) ( 1 1 2 ' z p z f D z f D z zp z p n n − = ⋅ +₊ .

From here we obtain:

). ( ) ( 1 ) ( ) ( ) ( ' 1 2 z zp z p z p z f D z f D n n ⋅ + = + +

Thus from (2) we obtain:

) 3 ( ) ( ) ( 1 ) ( ) ( ) ( ' 1 z zp z p z p z F D z F D n n ⋅ + = + .

If we consider h∈Hu(U), with h(0)=1, which maps the unit disc into the convex domain included in right half plane ∆_α,γ , then from

) ( ) ( 1 z F D z F D n n+

take all values in ∆_α,γ , using (3) we obtain:

) ( ) ( ) ( 1 )

( zp' z h z

z p z

4

where, from her construction, we have Reh(z)>0 and from theorem (2.1) we obtain )

( ) (z h z

p p . From here follows that

) ( ) ( 1 z f D z f D n n+

take all values in the convex domain included in right half plane ∆_α,γ , or f(z)=IF(z)∈US_n(α, γ), with α ≥0 and

0 > γ .

Theorem 3.2. If F(z)∈UCC_n(α, γ), in respect to the function n-uniform starlike of order

γ

and type α G(z), with α ≥0 and γ >0, then

) , ( ) ( )

(z IF z UCC_n α γ

f = ∈ in respect to the function n-uniform starlike of order

γ

and type

α

, see theorem (3.1), g(z)=IG(z) with

α

≥0 and

γ

>

0

.

Proof. We know that

F

(

z

)

∈

UCC

_n

(

α

,

γ

)

if and only if

) ( ) ( 1 z G D z F D n n+ take all values in the convex domain included in right half plane ∆_α,γ .

By differentiating (1) we obtain: ) ( ) (z zf' z

F = and G(z)=zg'(z) By means of the application of the linear operator

D

n+1 we obtain:

) ( ) ( 2 1 z f D z F

Dn+ = n+ and DnG(z)=Dn+1g(z). With simple calculation we obtain:

) 4 ( ) ( ) ( ) ( ) ( 1 2 1 z g D z f D z G D z F D n n n n + + + = .

With notation ( )

) ( ) ( 1 z p z g D z f D n n = +

, and ( )

) ( ) ( 1 z h z g D z g D n n = +

it follows that:

) ( ) ( 1 ) ( ) ( ) ( _' 1 2 z zp z h z p z g D z f D n n ⋅ + = + + . Thus from (4) we obtain:

) 5 ( ) ( ) ( 1 ) ( ) ( ) ( _' 1 z zp z h z p z G D z F D n n ⋅ + = + .

If we consider q convex in unit disc U, which maps the unit disc into the convex domain included in right half plane ∆_α,γ , then from

) ( ) ( 1 z G D z F D n n+

take all values in

γ α,

∆ , using (5) we obtain:

) ( ) ( ) ( 1 )

( zp' z q z

z h z

5

where, from her construction, we have Reh(z)>0. From here follows that 0

) ( 1 Re >

z

h . In this conditions from theorem (2.2) we obtain p(z)pq(z). From here

follows that

) (

) ( 1

z g D

z f D

n n+

take all values in the convex domain included in right half plane ∆_α,γ ; or f(z)=IF(z)∈UCCn(α, γ), in respect to

) , ( ) ( )

(z IG z USn α γ

g = ∈ with α ≥0 and γ >0. 3. Some particular cases

1. From theorem (3.1), for n=1, we obtain that the integral operator (1) preserved the class USc(α, γ), with

γ

>

0

, of uniform convex of type

α

and of order

γ

functions, introduced by I. Magdaş.

2. From theorem (3.1), for n=1,

γ

=

0

we obtain that the integral operator (1) preserved the class

US

c

(

α

)

, of uniform convex of type

α

functions, introduced by S. Kanas and A. Visniowska.

3. From theorem (3.1), for n=1,

γ

=

0

,

α

=

1

, we obtain that the integral operator (1) preserved the class

US

c, of uniform convex function, introduced by A. W. Goodman, and studied by W. Ma and D. Minda.

4. From theorem (3.1), for n=1,

α

=1, we obtain that the integral operator (1) preserved the class USc[γ], with

γ

>

0

, of uniform convex of order

γ

functions, introduced by F. Ronning.

5. From theorem (3.1), for n=0,

α

=1, we obtain that the integral operator (1)

preserved the class 

  



 − +

2 1 , 2

1 γ γ

SP , introduced by F. Ronning.

6. From theorem (3.2), for

γ

=

0

, we obtain that the integral operator (1) preserved the class UCCn(α), of n-uniform close to convex of type

α

in respect to a n-uniform starlike of type

α

function, introduced by D. Blezu.

REFERENCES

[1] D. Blezu “On the n-uniform close to convex function with respect to a convex domain”, Demontratio Mathematica (to appear).

[2] I. Magdaş, Doctoral thesis, University “Babeş-Bolyai” Cluj-Napoca 1999.

6

[4] S. S. Miller and P. T. Mocanu “Univalent solution of Briot-Buoquet differential equations”, J. Differential Equations 56 (1985), 297-308.

[5] S. S. Miller and P. T. Mocanu “On some classes of first-order differential subordinations”, Mich. Math. 32 (1985), 185-195.

[6] Gr. Sălăgean “On some classes of univalent functions” Seminar of geometric function theory Cluj-Napoca 1983.

Authors: