J. Nonlinear Sci. Appl. 2 (2009), no. 4, 219–224
The Journal of Nonlinear Sciences and Applications
http://www.tjnsa.com
CERTAIN SUBCLASSES OF CONVEX FUNCTIONS WITH
POSITIVE AND MISSING COEFFICIENTS BY USING A
FIXED POINT
SH. NAJAFZADEH1 , A. EBADIAN2 AND M. ESHAGHI GORDJI3∗
Abstract. By considering a fixed point in unit disk ∆, a new class of univalent convex functions is defined. Coefficient inequalities, integral operator and
extreme points of this class are obtained.
1. Introduction
Let w be fixed point in ∆ = {z : |z| < 1} and
A(w) = {f ∈ H(∆) : f (w) = f 0 (w) − 1 = 0}.
Let Mw = {f ∈ A(w) : f is Univalent in ∆}, and Mw denoted the subclass of
A(w) consist of the functions of the form
+∞
X
1
f (z) =
+
an (z − w)n , an ≥ 0, z 6= w
z − w n=k
(1.1)
The function f (z) in Mw is said to be Convex of order η(0 < η < 1) if and only if
(z − w)f 00 (z)
} > η , (z ∈ ∆).
f 0 (z)
The similar definitions for uniformly convex and starlike functions are introduced
by Goodman in [4, 5]. For the function f (z) in Mw , Ghanim and Darus have
defined an operator as follows:
Re{1 +
I 0 f (z) = f (z)
Date: Received: 2 March 2009; Revised: 15 June 2009.
∗
Corresponding author.
2000 Mathematics Subject Classification. Primary 30C45,30C50.
Key words and phrases. Subordination; P-valent function; Coefficient estimate; Fixed point;
Distortion bound; Convexity.
219
220
SH NAJAFZADEH, A. EBADIAN AND M. ESHAGHI GORDJI
2
z−w
2
I 2 f (z) = (z − w)(I 1 f (z))0 +
z−w
I 1 f (z) = (z − w)f 0 (z) +
and for m = 1, 2, ...
I m f (z) = (z − w)(I m−1 f (z))0 +
2
z−w
(1.2)
+∞
X
1
=
+
nm an (z − w)n .
z − w n=k
We note that various other subclasses have been studied rather extensively by
many authors (see [1, 2, 3] and [6]).
Definition 1.1. A function f (z) belonging to the class Mw is in the class Mw (A, B, m)
if it satisfies the condition
|B
1
[(z
2
00
m
− w) (I(I mff(z))
+ 1] +
(z))0
m
00
[(z − w) (I(I mff(z))
+ 1] +
2
(z))0
1
2
A+1
2
|≤1
(1.3)
for some −1 ≤ B < A < 1 , 0 ≤ A ≤ 1.
2. Main Results
First we obtain coefficient inequalities for functions in Mw (A, B, m). Then we
prove the linear combination property.
Theorem 2.1. The function f (z) of the form (1.1) belongs t Mw (A, B, m) if and
only if
+∞
X
nm+1 [n(B + 1) + A + 2]an ≤ 1 + A − B.
(2.1)
n=k
Proof. Suppose (2.1) holds and
1
B
B+A
H = | (z − w)(I m f (z))00 + (I m f (z))0 | − | (z − w)(I m f (z))00 +
| (2.2)
2
2
2
Replacing (I m f (z))0 and (I m f (z))00 by their series expansion for 0 < |z − w| =
r < 1, we have
H=|
+∞
X
1
n=k
1
1
nm+1 (n + 1)an (z − w)n−1 | − | (B − A − 1)
2
2
(z − w)2
+
≤|
+∞
X
1
n=k
2
+∞
X
1
nm+1 [ (Bn + A + 1)]an (z − w)n−1 |
2
n=k
+∞
n
m+1
(n + 1)an r
n−1
1
1 X m+1 1
− (1 + A − B) 2 +
n
[ (Bn + A + 1)]an rn−1 |
2
r
2
n=k
CERTAIN SUBCLASSES OF CONVEX FUNCTIONS
221
Since this inequality holds for all r (0 < r < 1), making r → 1, we have
H≤
+∞
X
1
n=k
1
nm+1 [n(B + 1) + A + 2]an − (1 + A − B)
2
2
by (2.1)
H ≤ 0.
So we have the required result.
Conversely, letP
1
n
f (z) = z−w
+ +∞
n=k an (z − w) and (1.3) holds, then we have
P+∞ 1 m+1
(n + 1)an (z − w)n−1
n=k 2 n
|1
| ≤ 1,
P+∞ m+1 1
1
(B − A − 1) (z−w)
[ 2 (Bn + A + 1)]an (z − w)n−1
2 +
n=k n
2
hence,
P+∞
|1
2
(1 + A +
1 m+1
(n + 1)an (z
n=k 2 n
P+∞ 1 m+1
B) − n=k 2 n
(Bn +
− w)n+1
A + 1)an (z − w)n+1
| ≤ 1.
Since Re(z) ≤ |z| for all z, then it follows from above inequality that
P+∞ 1 m+1
(n + 1)an (z − w)n+1
n=k 2 n
Re{ 1
} ≤ 1.
P
1 m+1
(1 + A + B) − +∞
(Bn + A + 1)an (z − w)n+1
n=k 2 n
2
By putting z − w = r with 0 < r < 1 in above inequality, we obtain
P+∞ 1 m+1
(n + 1)an rn+1
n=k 2 n
≤ 1.
P
1 m+1
1
(1 + A − B) − +∞
(Bn + A + 1)an rn+1
n=k 2 n
2
(2.3)
Upon clearing the denominator in (2.3) and letting r → 1, we get
+∞
X
1
n=k
2
n
m+1
+∞
X
1
1 m+1
(n + 1)an ≤ (1 + A − B) −
n
(Bn + A + 1)an ,
2
2
n=k
so
+∞
X
1
n=k
1
nm+1 [n(B + 1) + A + 2]an ≤ (1 + A − B).
2
2
This completes the proof.
Theorem 2.2. Let fj (z) defined by
+∞
fj (z) =
X
1
+
an,j (z − w)n , j = 1, 2, ...
z − w n=k
be in the class Mw (A, B, m), then the function
F (z) =
t
X
dj fj (z) , dj ≥ 0
j=1
is also in Mw (A, B, m), where
Pt
j=1
dj = 1.
(2.4)
222
SH NAJAFZADEH, A. EBADIAN AND M. ESHAGHI GORDJI
Proof. Since fj (z) ∈ Mw (A, B, m), by (2.1) we have,
+∞
X
nm+1 [n(B + 1) + A + 2]an,j ≤ 1 + A − B
, j = 1, 2, ...
(2.5)
n=k
also
F (z) =
t
X
+∞
dj (
j=1
X
1
+
an,j (z − w)n )
z − w n=k
t
+∞ X
t
X
1 X
dj +
(
dj an,j )(z − w)n
=
z − w j=1
n=k j=1
+∞
=
t
X
X
1
+
sn (z − w)n where sn =
dj an,j
z − w n=k
j=1
But
+∞
X
nm+1 [n(B + 1) + A + 2]sn =
n=k
+∞
X
nm+1 [n(B + 1) + A + 2][
t
X
j=1
n=k
=
t
X
dj {
j=1
+∞
X
dj an,j ]
nm+1 [n(B + 1) + A + 2]an,j }
n=k
by (2.5)
≤
t
X
dj (1 + A − B) = 1 + A − B.
j=1
Now the proof is complete.
3. Extreme points and Integral operators
In the last section we investigate about extreme points of Mw (A, B, m) and
verify the effect of two operators on functions in the class Mw (A, B, m).
Theorem 3.1. Let
f0 (z) =
1
z−w
and
1
1+A−B
+ m+1
(z − w)n , n ≥ k
(3.1)
z−w n
[n(B + 1) + A + 2]
then
the form f (z) =
P∞ f (z) ∈ Mw (A, B, m) if and only if it can be expressed
Pin
+∞
C
f
(z)
where
C
≥
0
,
C
=
0
(i
=
1,
2,
...,
k
−
1)
,
Cn = 1
n
i
n=1 n n
0
P
Proof. Let f (z) = +∞
Cn fn (z), so,
0
fn (z) =
k−1
X
C0
1
1+A−B
f (z) =
+
Cn [
+ m+1
(z − w)n ]+
z − w n=1
z−w n
[n(B + 1) + A + 2]
+∞
X
n=k
Cn [
1
1+A−B
+ m+1
(z − w)n ]
z−w n
[n(B + 1) + A + 2]
CERTAIN SUBCLASSES OF CONVEX FUNCTIONS
223
+∞
X
1+A−B
1
+
Cn (z − w)n .
=
m+1
z − w n=k n
[n(B + 1) + A + 2]
Since
+∞
X
n=k
1+A−B
nm+1 [n(B + 1) + A + 2]
Cn
nm+1 [n(B + 1) + A + 2]
1+A−B
=
+∞
X
Cn =
+∞
X
Cn = 1 − C0 ≤ 1,
n=1
n=k
so f (z) ∈ Mw (A, B, m).
Conversely , suppose that f (z) ∈ Mw (A, B, m). Then by (2.1) we have
1+A−B
.
0 ≤ an ≤ m+1
n
[n(B + 1) + A + 2]
By setting
nm+1 [n(B + 1) + A + 2]
Cn =
an , n ≥ 1
1+A−B
P+∞
Ci = 0 (i = 1, 2, ..., k − 1) , C0 = 1 − n=1 Cn , we obtain the required result. Theorem 3.2. Let γ be a real number such that γ > 1. If f (z) ∈ Mw (A, B, m),
then the functions
Z z
γ−1
(t − w)γ−1 f (t)dt and
H1 (z) =
(z − w)γ w
Z 1
ν C f (ν(z − w) + w)dν , C ≥ 1
H2 (z) = C
0
are also in the same class.
Proof. Let f (z) ∈ Mw (A, B, m), then a simple calculation shows that,
+∞
X 1
1
H1 (z) =
+
an (z − w)n
z − w n=k γ + n
+∞
X
1
C
H2 (z) =
+
an (z − w)n
z − w n=k C + n + 1
Since
+∞
X
C
C+n+1
n=k
1
γ+n
< 1, by using Theorem 2.1, we get,
+∞
nm+1 [n(B + 1) + A + 2]an
n=k
and
+∞
X
< 1 and
X
1
<
nm+1 [n(B + 1) + A + 2]an < 1 + A − B
γ + n n=k
+∞
n
m+1
X
C
[n(B +1)+A+2]an
<
nm+1 [n(B +1)+A+2]an < 1+A−B
C + n + 1 n=k
Hence by Theorem (2.1) we conclude that H1 (z) and H2 (z) are in the class
Mw (A, B, m). So the proof is complete.
224
SH NAJAFZADEH, A. EBADIAN AND M. ESHAGHI GORDJI
References
1. F. Ghanim and M. Darus, On new subclass of analytic univalent function with negative
coefficient I, Int. J. Contemp. Math. Sciences, 27 (2008) , 1317-1329. 1
2. ———————, Some subordination results associated with certain subclass of analytic
meromorphic function, J. Math. and statistics, 4 4(2008), 112-11. 1
3. F. Ghanim, M. Darus and S. Sivasubramanian, On subclass of analytic univalent function,
Int. J. pure Appl., 40 (2007), 307-319. 1
4. A .W. Goodman, On uniformly starlike functions, J. Math. Anal. Appl, 155 (1991), 364-370.
1
5. ———————– , On uniformaly convex functions, Ann. polon. Math, 56 (1991), 87-92. 1
6. Sh. Najafzadeh, A. Tehranchi and S. R. Kulkarni, Application of differential operator on
p-valent meromorphic functions, Anal. Univ. oradea, Fasc. Math. Tom XII, (2005), 75-90.
1
1
Department of Mathematics, Faculty of Science,, University of Maragheh,
Maragheh, Iran
E-mail address: najafzadeh1234@yahoo.ie
2
Department of Mathematics, Faculty of Science, Urmia University, Urmia,
Iran
E-mail address: ebadian.ali@gmail.com
3
Department of Mathematics, Semnan University, P. O. Box 35195-363, Semnan, Iran.
E-mail address: madjid.eshaghi@gmail.com