General Mathematics Vol. 12, No. 3 (2004), 61–66
Sufficient conditions for n-starlikeness
Viorel Cosmin Holhoş
Abstract
In this paper we obtain a sufficient condition for n-starlikeness of
the form:
α
Dn+1 f (z)
Dn+2 f (z)
+
(β
−
α)
≺ h (z)
Dn f (z)
Dn f (z)
where h (z) is an univalent function in the unit disc U and Dn is the
Sălăgean differential operator.
2000 Mathematical Subject Classification: 30C45
Keywords: Univalent functions, Sălăgean differential operator.
1
Introduction
Let An , n ∈ N∗ denote the class of functions of the form:
f (z) = z +
∞
X
ak z k
k=n+1
61
62
Viorel Cosmin Holhoş
which are analytic in the unit disc U = {z ; z ∈ C, |z| < 1} and A1 = A.
We note
½
¾
zf 0 (z)
S = f ∈ A : Re
> 0, z ∈ U
f (z)
∗
the class of functions f ∈ A which are starlike in the unit disc.
We denote by K the class of functions f ∈ A which are convex in the
unit disc U , that is
½
µ 00
¶
¾
zf (z)
+ 1 > 0, z ∈ U .
K = f ∈ A : Re
f 0 (z)
For f ∈ An we define the Sălăgean differential operator Dn by
D0 f (z) = f (z)
D1 f (z) = Df (z) = zf 0 (z)
and
Dn+1 f (z) = D (Dn f (z)) ;
n ∈ N∗ .
Let α ∈ [0, 1) and n ∈ N. The class Sn (α) named the class of n−starlike
function of order α is defined by
½
¾
Dn+1 f (z)
Sn (α) = f ∈ A : Re
> α, z ∈ U .
Dn f (z)
Theorem 1 (see [1]).
Let q be a univalent function in U
and let the
functions θ, φ be analytic in a domain D containing q(U ), with φ(w) 6= 0,
when w ∈ q(U ). Let Q(z) = zq 0 (z)φ (q(z)) , h(z) = θ (q(z)) + Q(z) and
suppose that
(i)
Q
is starlike in U
· 0
¸
zh0 (z)
θ (q (z)) zQ0 (z)
(ii) Re
= Re
+
> 0,
Q (z)
φ (q (z))
Q (z)
z ∈ U.
Sufficient conditions for n-starlikeness
63
If p is analytic in U , with p(0) = q(0) , p(U ) ⊂ D, and
θ (p(z)) + zp0 (z)φ (p(z)) ≺ θ (q(z)) + zq 0 (z)φ (q(z)) = h(z)
then p (z) ≺ q (z) and q is the best dominant.
2
Main results
Theorem 2. Let n ∈ N, let q be a convex function in U, with q (0) = 1,
(1)
Re q (z) >
and let f (z) ∈ A, with
α
α−β
, α > 0, α + β > 0
2α
f (z)
6= 0, that satisfy
z
Dn+2 f (z)
Dn+1 f (z)
+
(β
−
α)
≺ h (z) ,
Dn f (z)
Dn f (z)
where
h (z) = αzq 0 (z) + αq 2 (z) + (β − α) q (z)
then
Dn+1 f (z)
≺ q (z) , z ∈ U.
Dn f (z)
(2)
Proof. Let q be a convex function in U , with q (0) = 1 and in Theorem 1
we choose
θ (w) = αw2 + (β − α)w
φ (w) = α 6= 0
Q (z) = zq 0 (z) φ (q (z)) = αzq 0 (z)
and we have
(i) Q (z) = αzq 0 (z) is starlike in U, because q is convex
64
Viorel Cosmin Holhoş
· 0
¸
θ (q (z)) zQ0 (z)
zh0 (z)
= Re
+
(ii) Re
=
Q (z)
φ (q (z))
Q (z)
·
¸
2αq (z) + β − α zQ0 (z)
+
>0
= Re
α
Q (z)
because (1).
The conditions of Theorem 1 are satisfied and for p (z) = 1 + p1 z + ...
which satisfies:
αp2 (z) + (β − α) p (z) + αzp0 (z) ≺ h (z)
we have p (z) ≺ q (z) and q is the best dominant.
If we let
p (z) =
Dn+1 f (z)
Dn f (z)
then
Dn+2 f (z)
Dn+1 f (z)
+
(β
−
α)
≺
Dn f (z)
Dn f (z)
≺ αq 2 (z) + (β − α) q (z) + αzq 0 (z)
αp2 (z) + (β − α) p (z) + αzp0 (z) = α
which implies that
Dn+1 f (z)
≺ q (z) .
Dn f (z)
Remark 3. For n = 0 we obtain the result given in [2].
For n = 1 we have the following result:
Corollary 4. Let q be a convex function in U, with q (0) = 1,
(3)
Re q (z) >
α−β
, α > 0, α + β > 0.
2α
If
α
D3 f (z)
D2 f (z)
+ (β − α)
≺ αq 2 (z) + (β − α) q (z) + αzq 0 (z) ,
Df (z)
Df (z)
Sufficient conditions for n-starlikeness
65
then
D2 f (z)
≺ q (z) , z ∈ U.
Df (z)
(4)
1+z
then
1−z
¶2
µ
1+z
1+z
2z
+ (β − 1)
, z∈U
h (z) =
2 +
1−z
1−z
(1 − z)
If we let α = 1, β ≥ 1 and q (z) =
(5)
and from Theorem 2 we have:
Corollary 5. If f (z) ∈ A, with
f (z)
6= 0, satisfies
z
Dn+2 f (z)
Dn+1 f (z)
+
(β
−
1)
≺ h (z)
Dn f (z)
Dn f (z)
where h is given by (5), then
Dn+1 f (z)
1+z
≺
, z ∈ U.
n
D f (z)
1−z
(6)
The relation (6) is equivalent to
Dn+1 f (z)
Re n
>0
D f (z)
that is f is n-starlike function. In this case for n = 0 we obtain the following
starlikeness condition:
Corollary 6. If f (z) ∈ A, with
f (z)
6= 0, satisfies
z
D1 f (z)
D2 f (z)
+ (β − 1)
≺ h (z) ,
Df (z)
Df (z)
where h is given by (5), then
(7)
zf 0 (z)
> 0, z ∈ U.
Re
f (z)
66
Viorel Cosmin Holhoş
For n=1 we obtain the following convexity condition:
Corollary 7. If f (z) ∈ A, with
f (z)
6= 0, satisfies
z
D3 f (z)
D2 f (z)
+
(β
−
1)
≺ h (z) ,
D0 f (z)
D0 f (z)
where h is given by (5), then
¶
µ
zf 00 (z)
(8)
Re 1 + 0
> 0, z ∈ U.
f (z)
References
[1] S. S. Miller, P. T. Mocanu, On some classes of first-order differential
subordinations, Michigan Math. J., 32(1985), 185-195.
[2] P. T. Mocanu, G. Oros, Sufficient conditions for starlikeness II, Studia
Univ. Babeş-Bolyai, Math., 43,1(1998), 57-62.
[3] G. S. Sălăgean, Subclasses of univalent functions, Lecture Notes in
Math, Springer-Verlag, 1013(1983), 362-372.
Universitatea Ecologică “Traian”
Str. 1 Decembrie 1918, nr. 14
330018 Deva, Romania
E-mail: holhosv@yahoo.com