TWO STARLIKENESS PROPERTIES FOR THE BERNARDI
OPERATORS
by
Daniel Breaz and Nicoleta Breaz
Abstract. It is well known that if
f
is starlike, thenF
is also starlike.In this paper we extend this result iff
satisfies larger conditions and proving thatF
is starlike of order 2 3 and 13, whereF
is the Bernardi integral operator.Introductions.
Let U =
{
z: z <1}
the unit disc in the complex plane, and let H( )
U the space of holomorphic functions in U.Let An denote the class of functions
f
of the form( )
z z a z a z z Uf n n
n
n + + ∈
+
= +
+ +
+ 2 2 ...,
1 1
wich are analytic in the unit disc U and A1 = A.
Let the class of univalent functions S =
{
f :f ∈Hu( ) ( )
U ,f 0 = f′( )
0 −1=0}
and let( )
( )
( )
′ > ∈
=
∗ z U
z f
z f z f
S α :Re α,
be the class of the starlike functions of the order α,0≤α <1, in U. For α =0, denote S∗
( )
0 =S∗ the class of the starlike functions in U.Consider the following integral operator
( )
= +∫
z f( )
tt −dt ≥z z F
0
1 , 0
1 γ
γ γ
γ ,
where f ∈An.
Lemma A.[1] Let q univalent in U and let θ and φ analytic functions in the domain
( )
U qLet
( )
z nzq( ) ( )
z[ ]
q zQ = ′ φ h
( )
z =θ[ ] ( )
q( )
z +Q zand suppose that next condition is satisfy:
i) Q is starlike and
ii)
( )
( )
Re[ ]
[ ]
( )
( )
( )
( )
0Re >
′ + ′
= ′
z Q
z Q z z q
z q z
Q z h z
φ
θ .
If p is analytic in U, with
( ) ( ) ( )
q p p( )( )
p( )
U Dp 0 = 0, ′0 =...= n−1 0 =0, ⊂
and
( )
[ ]
p z +zp′( ) ( )
zφ[ ]
p z θ[ ]
q( )
z +zq′( ) ( )
zφ[ ] ( )
q z =h zθ p
then ppqand q is the best dominant.
Main results.
Theorem 1. Let
( )
( )(
z nz z)
z z
h
γ γ − + − + − =
2 2 2
2 2
2
If f ∈An and
( )
( ) ( )
h zz f
z f z
p
′ ,
then
( )
( )
32 Re ′ >
z F
z F z
,
where
Proof. The relation (1) implies:
( )
z zF( ) ( ) ( )
z f zF + ′ = γ +1
γ (2)
Let
( )
zFF( )
( )
zz zp = ′ .
The relation (2) implies
( )
( )
( )
f( )
( )
z z f z z p zp z p
z ′
= + + ′
γ .
But
( )
( ) ( )
h zz f
z f z
p
′
implies
( )
( )
p( ) ( )
z h z zp z p z
p
+ + ′
γ .
We apply the lemma A for proved that
( )
( )
32 Re ′ >
z F
z F z
.
If we let:
( )
z zq
− =
2 2
( )
w =wθ
( )
γφ
+ =
w
w 1
( )
[ ]
q z z− =
2 2 θ
( )
[ ]
z zz q
γ γ φ
− + − =
2 2
( )
( ) ( )
[ ] ( )(
z nz z)
z q z q nz z Q γ γ φ − + − = ′ = 2 2 2 2 .( )
[ ] ( )
( )
( )(
z nz z)
z z Q z q z h γ γ θ − + − + − = + = 2 2 2 2 2 2 .
Since Q is starlike and Reφ
[ ]
q( )
z >0, by lemma A we deduce( )
( )
( )
( )
32 Re 2 2 > ′ ⇒ − ′ ⇔ z F z F z z z F z F z q
pp p .
Theorem 2. Let
( )
( )
z(
nz( )
z)
z z z h γ γ + − + − + − + = 1 2 2 2 4 2 2 .
If f ∈An and
( )
( ) ( )
h zz f z f z p ′ , then
( )
( )
31 Re ′ >
z F z F z , where
( )
= +∫
z f( )
tt −dt z z F 0 1 1 γ γγ .Proof. We are
( )
z zF( ) ( ) ( )
z f zF + ′ = γ +1
γ (3)
Let
( )
zFF( )
( )
zz zp = ′ .
We deduce
( )
( )
( )
f( )
( )
z z f z z p z p z p z ′ = + + ′ γ . But( )
( ) ( )
z h z fz
p
and obtained
( )
( )
p( ) ( )
z h zz p z p z p + + ′ γ .
In lemma we consider:
( )
zzz q − + = 2 2
( )
w =wθ
( )
γ φ + = w w 1( )
[ ]
zzz q − + = 2 2 θ
( )
[ ]
( )
z zz q γ γ φ − + + − = 1 2 2 2 .
( )
( ) ( )
[ ] ( )
z(
nz( )
z)
z q z q nz z Q γ γ φ − + + − = ′ = 1 2 2 2 2
( )
[ ] ( )
( )
( )
z(
nz( )
z)
z z z Q z q z h γ γ θ − + + − + − + = + = 1 2 2 2 2 2 2 .
Sience Q is starlike and Reφ
[ ]
q( )
z >0, by lemma A obtained( )
( )
( )
( )
31 Re 1 1 > ′ ⇒ − + ′ ⇔ z F z F z z z z F z F z q
pp p
REFERENCES
[1].Miller, S.S. and Mocanu P.T., On some classes of first differential subordinations, Michigan Math. J. 32, pp. 185-195, 1985.
[3].Oros, Gh. On starlike images by an integral operator, Mathematica, Tome 42(65), No 1, 2000, pp. 71-74.
[4].Breaz, D., Breaz, N. Starlikeness conditions for the Bernardi operator, to be appear.
Authors:
Daniel Breaz- „1 Decembrie 1918” University of Alba Iulia, str. N. Iorga, No. 13, Departament of Mathematics and Computer Science, email: [email protected]
