General Mathematics Vol. 17, No. 4 (2009), 149–156
Application of Differential subordination on
p-valent Functions with a fixed point
Sh. Najafzadeh and A. Rahimi
Abstract
Making Use of the familiar differential subordination Structure
in this paper, we investigate a new class of p-valent functions with
a fixed point w. Some results connected to sharp coefficient bounds,
Distortion Theorem and other important properties are obtained.
2000 Mathematics Subject Classification:30C45,30C50
Key words and phrases: Subordination, P-valent function,Coefficient
estimate, Distortion bound and Radii of starlikeness and Convexity.
1
Introduction
Let Sw (P ) for a fixed point w denote the class of functions f (z) of the form
+∞
X
1
+
an+p (z − w)n+p
f (z) =
(z − w)p n=1
149
an+p ≥ 0 (1.1)
150
Sh.Najafzadeh and A.Rahimi
For the function f (z) in the class Sw (P ), Frasin and Darus [3] defined the
differential operator Dk as follow:
D0 f (z) = f (z)
D1 f (z) = (z − w)f´(z) +
1+p
(z − w)p
D2 f (z) = (z − w)(D1 f (z))0 +
1+p
(z − w)p
and for k = 1, 2, 3...
Dk f (z) = (z − w)(Dk−1 f (z))0 +
p+1
(z − w)p
+∞
X
1
=
+
(n + p)k an+p (z − w)n+p
p
(z − w)
n=1
(1.2)
See also [4].
Now we define the class Ωw p(A, B) consisting the functions f (z)²Sw (p)
such that
−
(z − w)(Dk f (z))00
≺ H(z) (1.3)
[Dk f (z)]0
1+Az
where H(z) = (p + 1) 1+Bz
, A = B + (C − B)(1 − λ), −1 ≤ B < C ≤ 1, 0 ≤
λ < 1 and ” ≺ ” denotes the subrdination Symbol. See [2],[5] and [1].
2
Main Results
In this section we find sharp Coefficient estimates and Integral representation for the class Ωw
p (A, B).
Application of Differential subordination
151
Theorem 2.1 Let f (z) ∈ Sw (p), then f (z) ∈ Ωw
p (A, B) if and only if
+∞
X
[(n+2p)(B+1)+(p+1)(C−B)(1−λ)]an+p < P (p+1)(C−B)(1−λ). (2.1)
n=1
The result is sharp for the function h(z) given by
h(z) =
1
P (p + 1)(C + B)(1 − λ)
+
(z−w)n , n = 1, 2, ... (2.2)
p
(z − w) (n + 2p)(B + 1) + (p + 1)(C + B)(1λ)
proof. Let f (z) ∈ Ωw
p (A, B), then the inequality (1.2) or inequality
|
(z − w)(Dk f (z))00 + (p + 1)0 (Dk f (z))0
| < 1 (2.3)
B(z − w)(Dk f (z))00 + (p + 1)[B + (C − B)(1 − λ)](Dk f (z))0
holds true, therefore by using (1.1) we have
P+∞
(n + 2p)(n + p)k+1 an+p (z − w)n+p−1
| n=1
|<1
−p(p + 1)(C − B)(1 − λ) + A
where
A=
+∞
X
[(p +1)(B +(C − B)(1 − λ))+ B(n + p+1)](n+ p)k an+p (z −w)n+p−1 .
n=1
Since Re(z) ≤ (z) for all z, therefore
P+∞
(n + 2p)(n + p)k+1 an+p (z − w)n+p−1
Re{ n=1
} < 1.
p(p + 1)(C − B)(1 − λ) − A
where
A=
+∞
X
[(p + 1)(B + (C − B)(1 − λ)) + B(n + p + 1)](n + p)k an+p (z − w)n+p−1
n=1
By letting (z − w) → 1̄ through real valves, we have
+∞
X
[(n + 2p)(B + 1) + (p + 1)(C − B)(1 − λ)]an+p < p(p + 1)(C − B)(1 − λ).
n=1
152
Sh.Najafzadeh and A.Rahimi
Conversely, Let (2.1) holds true, if we Let (z−w)∂∂44∗ where ∂4∗ denotes
the boundary of 4∗ , then we have
|
(z − w)(Dk f (z))00 + (p + 1)(Dk f (z))0
|≤
B(z − w)(Dk f (z))00 + (p + 1)[B + (C − B)(1 − λ)](Dk f (z))0
P+∞
k+1
|an+p |
n=1 (n + 2p)(n + p)
P+∞
P (p + 1)(C − B)(1 − λ) − n=1 [B(n + 2p) + (p + 1)(C + B)(1 + λ)]|an+p |
(by(2.1))
<1
Thus by the Maximum modolus Theorem, we conclude f (z) ∈ Ωw
p (A, B).
Theorem 2.2 If f (z) ∈ Ωw
p (A, B), then
Z z
Z z
(p + 1)(δM (t) − 1)
k
dt]ds
[exp
D f (z) =
0 (t − w)[1 − BM (t)
0
(2.4)
where δ = B + (c − B)(1 − λ) and |M (z)| < 1.
proof. Since f (z) ∈ Ωw
p (A, B), so (1.2) or equivalently (2.3) holds true.
Hence
(z − w)(Dk f (z))00 + (p + 1)(Dk f (z))0
= M (z),
B(z − w)(Dk f (z))00 + (p + 1)δ(Dk f (z))0
where |M (z)| < 1, z²4∗ and δ = B + (C − B)(1 − λ).
This yields
[Dk f (z)]00
(p + 1)[δM (z) − 1]
=
.
[Dk f (z)]0
(z − w)(1 − BM (z))
After integration we obtain the required result.
Remark. Theorem 2.1 shows that if f (z)²Ωw
p (A, B), then
|an+p | ≤
p(p + 1)(C − B)(1 − λ)
, n = 1, 2, 3, ... (2.5)
(2p + 1)(B + 1) + (p + 1)(C − B)(1 − λ)
Application of Differential subordination
3
153
Distortion Bounds and Extreme points
In this section we investigate about Distortion theorem and extreme points
of the class Ωw
p (A, B).
Theorem 3.1 Let f (z)²Ωw
p (A, B), then
r−p −
p(p + 1)(c − B)(1 − λ)
rp+1 < |Dk f (z)|
(2p + 1)(B + 1) + (p + 1)(C − B)(1 − λ)
< r−p +
p(p + 1)(C − B)(1 − λ)
rp+1
(2p + 1)(B + 1) + (p + 1)(C − B)(1 − λ)
(3.1)
where 0 < |z − w| = r < 1.
proof. By theorem 2.1 and (2.5) we have
+∞
X
1
+
(n + p)k an+p (z − w)n+p |
|D f (z)| = |
(z − w)p n=1
k
≤r
−p
+
+∞
X
(n + p)k |an+p |rn+p
n=1
< r−p +
p(p + 1)(C − B)(1 − λ)
rp+1 .
(2p + 1)(B + 1) + (p + 1)(C − B)(1 − λ)
similarly we obtain
|Dk f (z)| ≥ r−p −
p(p + 1)(C − B)(1 − λ)
rp+1 .
(2p + 1)(B + 1) + (p + 1)(C − B)(1 − λ)
So the proof is complete.
Theorem 3.2 The function f (z) of the from (1.1) belongs to Ωw
p (A, B) if
and only if it can be expressed by
f (z) =
+∞
X
n=p
γn fn (z), γn ≥ 0,
n = p, p + 1, ... (3.2)
154
Sh.Najafzadeh and A.Rahimi
where fp (z) =
fn (z) =
1
(z−w)p
(3.2),
p(p + 1)(C − B)(1 − λ)
1
+
(z−w)n , n = p+1, p+2, ...
(z − w)p (n + 2p) + (B + 1) + (p + 1)(C − B)(1 − λ)
+∞
X
γn = 1.
n=p
proof. Let
f (z) =
+∞
X
γn fn (z) = γp fp (z)+
n=p
+∞
X
γn [
n=p+1
p(p + 1)(C − B)(1 − λ)
1
+
(z − w)n ]
p
(z − w)
(n + 2p)(B + 1) + (p + 1)(C − B)(1 − λ)
+∞
X
1
p(p + 1)(C − B)(1 − λ)
=
+
γn (z − w)n .
p
(z − w)
(n
+
2p)(B
+
1)
+
(p
+
1)(C
−
B)(1
−
λ)
n=p
Now by using Theorem 2.1 we conclude that f (z) ∈ Ωw
p (A, B). Conversely,
P+∞
if f (z) given by (1.1) belongs to Ωw
p (A, B). by letting γp = 1 −
n=p+1 γn
where
γn =
(n + 2p)(B + 1) + (p + 1)(C − B)(1 − λ)
an+p , n = 1, 2, ...
p(p + 1)(C − B)(1 − λ)
we conclude the required result.
4
Radii of starlikeness and convexity
In the last section we introduce the Radii of Starlikeness and convexity for
functions in the class Ωw
p (A, B).
Application of Differential subordination
155
Theorem 4.1 If f (z) ∈ Ωw
p (A, B), then f is starlike of order ξ(0 ≤ ξ < p)
in disk |z − w| < R, and it is convex of order ξ in disk |zw| < R2 where
R1 = infn
[(2p + 1)(B + 1) + (p + 1)(C − B)?(1 − λ)](p − ξ) 1
(n + 3p + ξ)p(p + 1)(C − B)(1 − λ)
n+p
R2 = infn
(n + p)[(2p + 1)(B + 1) + (p + 1)(C − B)(1 − λ)] 1
(n + 3p + ξ)(p + 1)(C − B)(1 − λ)
n+p
proof. For Starlikeness it is enough to show that
|
but
(z − w)f 0 (z)
+ p| < pξ ,
f (z)
P+∞
n+p
(n + 2p)an+p (z − w)n+p
(z − w)f 0 (z)
|
+ p| = | n=1
|≤
P+∞
1
f (z)
+
a
(z
−
w
n+p
p
n=1
(z−w)
P+∞
n+p
n=1 (n + 2p)an+p |z − w|
≥p−ξ
P+∞
1 − n=1 an+p |z − w|n+p
or
+∞
X
(n + 2p)an+p |z − w|n+p ≤ p − ξ − (p − ξ)
+∞
X
an+p |z − w|n+p
n=1
n=1
+∞
X
n=1
(n + 3p + ξ)
an+p |z − w|n+p ≤ 1
p−ξ
by using (2.5) we obtain
+∞
X
(n + 3p + ξ)
n=1
+∞
X
n=1
p−ξ
an+p |z − w|n+p ≤
(n + 3p + ξ)P (P + 1)(C − B)(1 − λ)
|z − w|n+p ≤ 1
[(2P + 1)(B + 1) + (P + 1)(C − B)(1 − λ)](p − ξ)
So it is enough to suppose
|z − w|n+p ≤
[(2P + 1)(B + 1) + (P + 1)(C − B)(1 − λ)](p − ξ)
.
(n + 3p + ξ)P (P + 1)(C − B)(1 − λ)
For convexity by using the fact that ” f (z) is convex if and only if zf 0 (z) is
starlike ” and by an easy calculation we conclude the required result.
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Sh.Najafzadeh and A.Rahimi
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Shahram Najafzadeh
Department of Mathematics
University of Maragheh
Maragheh, Iran.
e-mail: najafzadeh1234@yahoo.ie