General Mathematics Vol. 14, No. 1 (2006), 55–64
On a subclass of n-uniformly close to convex
functions 1
Mugur Acu
Abstract
In this paper we define a subclass on n-uniformly close to convex
functions and we obtain some properties regarding this class.
2000 Mathematical Subject Classification: 30C45.
Key words and phrases: Uniformly close to convex functions,
Libera-Pascu integral operator, Briot-Bouquet differential subordination
1
Introduction
Let H(U ) be the set of functions which are regular in the unit disc
U = {z ∈ C : |z| < 1}, A = {f ∈ H(U ) : f (0) = f 0 (0) − 1 = 0}
and S = {f ∈ A : f is univalent in U }.
We recall here the definition of the well - known class of starlike functions:
zf 0 (z)
∗
S = f ∈ A : Re
>0, z∈U ,
f (z)
Let Dn be the Sălăgean differential operator (see [5]) Dn : A → A,
n ∈ N, defined as:
D0 f (z) = f (z)
1
Received January 10, 2006
Accepted for publication (in revised form) February 2, 2006
55
56
M. Acu
D1 f (z) = Df (z) = zf 0 (z)
Dn f (z) = D(Dn−1 f (z))
Remark 1.1. If f
∈
S , f (z)
=
z +
∞
P
aj z j , z
∈
U then
j=2
n
D f (z) = z +
∞
X
j n aj z j .
j=2
Let consider the Libera-Pascu integral operator La : A → A defined as:
(1)
1+a
f (z) = La F (z) =
za
Zz
F (t) · ta−1 dt ,
a∈C,
Re a ≥ 0.
0
For a = 1 we obtain the Libera integral operator, for a = 0 we obtain
the Alexander integral operator and in the case a = 1, 2, 3, ... we obtain the
Bernardi integral operator.
The purpose of this note is to define, using the Sălăgean differential
operator, a subclass on n-uniformly close to convex functions and to obtain
some properties regarding this class.
2
Preliminary results
Let k ∈ [0, ∞), n ∈ N∗ . We define the class (k, n) − S ∗ (see the definition
of the class (k, n) − ST in [1]) by f ∈ S ∗ and
n
n
D f (z)
D f (z)
Re
> k − 1 , z ∈ U .
f (z)
f (z)
Remark 2.1. (for more details see [1]) We denote by pk , k ∈ [0, ∞) the
function which maps the unit disk conformally onto the region Ωk , such that
1 ∈ Ωk and
∂Ωk = u + iv : u2 = k 2 (u − 1)2 + k 2 v 2 .
The domain Ωk is elliptic for k > 1, hyperbolic when 0 < k < 1, parabolic
for k = 1, and a right half-plane when k = 0. In this conditions, a function
On a subclass of n-uniformly close to convex functions
57
Dn f (z)
Dn f (z)
≺ pk or
take all
f (z)
f (z)
values in the domain Ωk . Because the domain Ωk is convex, as an immediate
consequence of the well known Rogosinski result for subordinate functions,
we obtain for p ≺ pk , p(z) = 1 + p1 z + p2 z 2 + ... , z ∈ U ,

8(arccos k)2


, 0 ≤ k < 1,


π 2 (1 − k 2 )


8
|pn | ≤ |P1 | := P1 (k) =
, k = 1,

π2

2

π


 √ 2
, k > 1.
4 κ(k − 1)K 2 (κ)(1 + κ)
f is in the class (k, n) − S ∗ if and only if
for n = 1, 2, ... , where K(κ) is Legendre, s complete elliptic integral of the
first kind, κ is chosen such that k = cosh[πK 0 (κ)]/[4K(κ)] and K 0 (κ) is
complementary integral of K(κ) .
With the notations from Remark 2.1 we have:
Theorem 2.1. [1] Let k ∈ [0, ∞) and f (z) = z +
∞
P
aj z j belongs to the
j=2
P1
class (k, n) − S ∗ . Then |a2 | ≤ n
and
2 −1
j−1 P1 Y
P1
|aj | ≤ n
1+ n
, j = 3, 4, ... , n ∈ N∗ .
j − 1 s=2
s −1
The next theorem is result of the so called ”admissible functions method”
introduced by P.T. Mocanu and S.S. Miller (see [2], [3], [4]).
Theorem 2.2. Let q be convex in U and j : U → C with Re[j(z)] > 0,
z ∈ U . If p ∈ H(U ) and satisfied p(z)+j(z)·zp0 (z) ≺ q(z), then p(z) ≺ q(z).
3
Main results
Definition 3.1. Let f ∈ A, k ∈ [0, ∞) and n ∈ N∗ . We say that the function f is in the class (k, n) − CC with respect to the function
g ∈ (k, n) − S ∗ if
n
n
D f (z)
D f (z)
Re
− 1 , z ∈ U .
> k · g(z)
g(z)
58
M. Acu
Remark 3.1. Geometric interpretation: f ∈ (k, n) − CC with respect to
Dn f (z)
≺ pk (see Remark 2.1)
the function g ∈ (k, n) − S ∗ if and only if
g(z)
Dn f (z)
or
take all values in the domain Ωk (see Remark 2.1).
g(z)
Remark 3.2. From the geometric properties of the domains Ωk we have
that
(k1 , n) − CC ⊂ (k2 , n) − CC, where k1 ≥ k2 .
Theorem 3.1. If F (z) ∈ (k, n) − S ∗ , with k ∈ [0, ∞) and n ∈ N∗ , then
f (z) = La F (z) ∈ (k, n) − S ∗ , where La is the integral operator defined by
(1).
Proof. By differentiating (1) we obtain
(2)
(1 + a)F (z) = af (z) + zf 0 (z) .
By means of the application of the linear operator Dn we have
(3)
(1 + a)Dn F (z) = aDn f (z) + Dn+1 f (z) .
From (2) and (3) we obtain
Dn f (z) Dn+1 f (z)
f
(z)
a
+
(1 + a)Dn F (z)
aDn f (z) + Dn+1 f (z)
f (z)
f (z)
=
=
0
0
zf (z)
(1 + a)F (z)
af (z) + zf (z)
f (z) a +
f (z)
or
Dn f (z) Dn+1 f (z)
+
Dn F (z)
f (z)
f (z)
.
=
0
zf (z)
F (z)
a+
f (z)
a
(4)
With notation p(z) =
zp0 (z) = z
Dn f (z)
, where p(0) = 1, we obtain
f (z)
(Dn f (z))0 f (z) − (Dn f (z)) f 0 (z)
=
f 2 (z)
On a subclass of n-uniformly close to convex functions
=
or
59
z (Dn f (z))0 Dn f (z) zf 0 (z)
Dn+1 f (z)
zf 0 (z)
−
·
=
− p(z) ·
f (z)
f (z)
f (z)
f (z)
f (z)
Dn+1 f (z)
zf 0 (z)
= zp0 (z) + p(z) ·
.
f (z)
f (z)
(5)
From (4) and (5) we have
zf 0 (z)
+ zp0 (z)
p(z) a +
Dn F (z)
f (z)
=
zf 0 (z)
F (z)
a+
f (z)
or
Dn F (z)
= p(z) +
F (z)
(6)
1
· zp0 (z)
zf 0 (z)
a+
f (z)
Dn F (z)
≺ pk (z), where pk maps the unit disk
F (z)
conformally onto the convex domain Ωk (see Remark 2.1).
1
Using (6) we obtain p(z) +
· zp0 (z) ≺ pk (z).
0
zf (z)
a+
f (z)
Using the hypothesis, from Theorem 2.2, we have p(z) ≺ pk (z) or
Dn f (z)
take all values in the domain Ωk . This means that f (z) ∈ (k, n)−S ∗ .
f (z)
From hypothesis we have
Theorem 3.2. If F (z) ∈ (k, n) − CC, k ∈ [0, ∞), n ∈ N∗ , with respect
to the function G(z) ∈ (k, n) − S ∗ , and f (z) = La F (z), g(z) = La G(z),
where La is the integral operator defined by (1), then f (z) ∈ (k, n) − CC,
k ∈ [0, ∞), n ∈ N∗ , with respect to the function g(z) ∈ (k, n) − S ∗ .
Proof. Using (1) and the linear operator Dn we obtain
(1 + a)Dn F (z) = aDn f (z) + Dn+1 f (z)
and
(1 + a)G(z) = ag(z) + zg 0 (z) .
60
M. Acu
From the above we have
(1 + a)Dn F (z)
aDn f (z) + Dn+1 f (z)
=
(1 + a)G(z)
ag(z) + zg 0 (z)
or
Dn f (z) Dn+1 f (z)
+
a
Dn F (z)
g(z)
g(z)
=
0
zg (z)
G(z)
a+
g(z)
n
D f (z)
If we denote p(z) =
, with p(0) = 1, we have
g(z)
n
D F (z)
=
G(z)
(7)
Dn+1 f (z)
g(z)
0
zg (z)
a+
g(z)
ap(z) +
With simple calculations we obtain
z (Dn f (z))0 Dn f (z) zg 0 (z)
Dn+1 f (z)
zg 0 (z)
zp (z) =
−
·
=
− p(z) ·
g(z)
g(z)
g(z)
g(z)
g(z)
0
and thus
(8)
(9)
Dn+1 f (z)
zg 0 (z)
= zp0 (z) + p(z) ·
g(z)
g(z)
From (7) and (8) we obtain
Dn F (z)
= p(z) +
G(z)
1
· zp0 (z) = p(z) + j(z) · zp0 (z) ,
zg 0 (z)
a+
g(z)
where from the hypothesis and the Theorem 3.1 we have Re j(z) > 0
z∈U.
From F (z) ∈ (k, n)−CC with respect to the function G(z) ∈ (k, n)−S ∗ ,
using Remark 3.1, we obtain p(z) + j(z) · zp0 (z) ≺ pk (z), where pk maps the
unit disk conformally onto the convex domain Ωk (see Remark 2.1).
Dn f (z)
From Theorem 2.2, we have p(z) ≺ pk (z) or
take all values in
g(z)
the domain Ωk . This means that f (z) ∈ (k, n) − CC with respect to the
function g(z) ∈ (k, n) − S ∗ .
On a subclass of n-uniformly close to convex functions
Theorem 3.3. If f (z) = z +
∞
X
61
aj z j belong to the class (k, n) − CC, with
j=2
∗
respect to the function g(z) ∈ (k, n) − S , g(z) = z +
∞
X
bj z j , where
j=2
∗
k ∈ [0, ∞), n ∈ N , then
|a2 | ≤
P1
P1 (P1 − 1 + 2n )
;
|a
|
≤
;
3
2n − 1
(2n − 1) (3n − 1)
j−1
Y
P1
P 1 − 1 + tn
|aj | ≤ n
·
, j ≥ 4,
j − 1 t=2 tn − 1
where P1 is given in Remark 2.1.
Dn f (z)
≺ pk (z),
g(z)
where pk (U ) = Ωk (see Remark 3.1). Let h(z) = 1 + c1 z + c2 z 2 + · · · ,
z ∈ U . Taking account the Rogosinski subordination theorem, we have
|cj | ≤ P1 , j ≥ 1 .
Using the hypothesis and the Remark 1.1 we have
Proof. We have f (z) ∈ (k, n) − CC if and only if h(z) =
z+
∞
X
j n aj z j
j=2
∞
X
z+
= 1 + c1 z + c2 z 2 + · · · .
bj z j
j=2
From the equality of the powers coefficients we obtain
2n a2 = c1 + b2 ; 3n a3 = c2 + b3 + c1 b2
and
(10)
j n aj = cj−1 + c1 bj−1 + c2 bj−2 + c3 bj−3 + · · · cj−2 b2 + bj , j ≥ 4 .
Using |cj | ≤ P1 , j ≥ 1 , 2n a2 = c1 + b2 and Theorem 2.1 we have
P1
2n
2 |a2 | ≤ P1 + n
= n
· P1
2 −1
2 −1
n
62
M. Acu
and thus |a2 | ≤
P1
.
−1
2n
P1 (P1 − 1 + 2n )
In a similarly way we obtain |a3 | ≤ n
.
(2 − 1) (3n − 1)
Using |cj | ≤ P1 , j ≥ 1 and Theorem 2.1 we obtain from (10) the estimations
"
(
#)
j−1
l−1 X
Y
P
P1
P
P
1
1
1
+ n
+
·
·
j n |aj | ≤ P1 1 + n
1+ n
n
2 − 1 l=3 l − 1 s=2
s −1
j −1
·
j−1 Y
t=2
P1
1+ n
t −1
.
By mathematical induction for j ≥ 4 we have
"
# Y
j−1
j−1
l−1 X
Y
P1
P1
P1
P1 − 1 + tn
1+ n
+
·
1+ n
=
2 − 1 l=3 ln − 1 s=2
s −1
tn − 1
t=2
and thus we obtain
n
j |aj | ≤ P1 ·
j−1
Y
P1 − 1 + tn
tn − 1
t=2
or
j−1 Y
P1
P1
+ n
·
1+ n
j − 1 t=2
t −1
j−1
Y
P1
P1 − 1 + tn
j |aj | ≤ j n
·
, j ≥ 4.
j − 1 t=2 tn − 1
n
Thus
n
j−1
Y
P1
P 1 − 1 + tn
|aj | ≤ n
·
, j ≥ 4,
j − 1 t=2 tn − 1
Theorem 3.4. Let a ∈ C, Re a ≥ 0, n ∈ N∗ and k ∈ [0, ∞). If
∞
X
F (z) ∈ (k, n) − CC, F (z) = z +
aj z j , and f (z) = La F (z),
f (z) = z +
∞
X
j=2
bj z j ,where La is the integral operator defined by (1), then
j=2
a + 1 P1 (P1 − 1 + 2n )
a + 1 P1
; |b3 | ≤ ;
|b2 | ≤ a + 2 2n − 1
a + 3 (2n − 1) (3n − 1)
On a subclass of n-uniformly close to convex functions
63
j−1
Y
a + 1 P1
P1 − 1 + tn
|bj | ≤ ·
, j ≥ 4,
a + j j n − 1 t=2 tn − 1
where P1 is given in Remark 2.1.
Proof. From f (z) = La F (z) we have
(1 + a)F (z) = af (z) + zf 0 (z) .
Using the above series expansions we obtain
(1 + a)z +
∞
X
(1 + a)aj z j = az +
j=2
∞
X
j=2
abj z j + z +
∞
X
jbj z j
j=2
and thus bj (a + j) = (1 + a)aj j ≥ 2 . a + 1
· |aj | , j ≥ 2 . Using the estimations
From the above we have bj ≤ a + j
from Theorem 3.3 we obtain the needed results.
For a = 1, when the integral operator La become the Libera integral operator, we obtain from the above theorem:
Corollary 3.1.Let n ∈ N∗ and k ∈ [0, ∞). If F (z) ∈ (k, n) − CC,
∞
∞
X
X
F (z) = z +
aj z j , and f (z) = L(F (z)), f (z) = z +
bj z j ,where L is
j=2
Z j=2
2 z
F (t)dt, then
the Libera integral operator defined by L(F (z)) =
z 0
|b2 | ≤
2 P1
1 P1 (P1 − 1 + 2n )
;
|b
|
≤
;
3
3 2n − 1
2 (2n − 1) (3n − 1)
j−1
Y
2
P 1 − 1 + tn
P1
|bj | ≤
·
, j ≥ 4,
j + 1 j n − 1 t=2 tn − 1
where P1 is given in Remark 2.1.
Remark 3.3. Similarly results with the results from the Corollary 3.1 are
easy to obtain from Theorem 3.4 by taking a = 0, respectively a = 1, 2, 3, · · · .
64
M. Acu
References
[1] S. Kanas, T. Yaguchi, Subclasses of k-uniformly convex and strlike functions defined by generalized derivate I, Indian J. Pure and Appl. Math.
32, 9(2001), 1275-1282.
[2] S. S. Miller and P. T. Mocanu, Differential subordonations and univalent
functions, Mich. Math. 28 (1981), 157 - 171.
[3] S. S. Miller and P. T. Mocanu, Univalent solution of Briot-Bouquet
differential equations, J. Differential Equations 56 (1985), 297 - 308.
[4] S. S. Miller and P. T. Mocanu, On some classes of first-order differential
subordinations, Mich. Math. 32(1985), 185 - 195.
[5] Gr. Sălăgean, Subclasses of univalent functions, Complex Analysis. Fifth
Roumanian-Finnish Seminar, Lectures Notes in Mathematics, 1013,
Springer-Verlag, 1983, 362-372.
”Lucian Blaga” University of Sibiu,
Department of Mathematics,
Str. Dr. I. Rat.iu, No. 5-7,
550012 - Sibiu, Romania
E-mail: acu mugur@yahoo.com