COEFFICIENT BOUNDS FOR MEROMORPHIC
STARLIKE AND CONVEX FUNCTIONS
SEE KEONG LEE
V. RAVICHANDRAN
Universiti Sains Malaysia
11800 USM Penang,
Malaysia
EMail: sklee@cs.usm.my
Department of Mathematics
University of Delhi
Delhi 110 007, India
EMail: vravi@maths.du.ac.in
Coefficient Bounds
S. K. Lee, V. Ravichandran
and S. Shamani
vol. 10, iss. 3, art. 71, 2009
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SUPRAMANIAM SHAMANI
School of Mathematical Sciences
Universiti Sains Malaysia
11800 USM Penang, Malaysia
EMail: sham105@hotmail.com
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Received:
30 January, 2008
Accepted:
03 May, 2009
Communicated by:
S.S. Dragomir
2000 AMS Sub. Class.:
Primary 30C45, Secondary 30C80.
Key words:
Univalent meromorphic functions; starlike function, convex function, FeketeSzegö inequality.
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Abstract:
Acknowledgment:
In this paper, some subclasses of meromorphic univalent functions in the
unit disk ∆ are extended. Let U (p) denote the class of normalized univalent meromorphic functions f in ∆ with a simple pole at z = p > 0. Let
φ be a function with positive real part on ∆ with φ(0) = 1, φ0 (0) > 0
which maps ∆ onto a region starlike with
P∗respect to 1 which is symmetric
with respect to the real axis. The class
(p, w0 , φ) consists of functions
f ∈ U (p) satisfying
p
pz
zf 0 (z)
−
+
−
≺ φ(z).
f (z) − w0
z − p 1 − pz
P
The class (p, φ) consists of functions f ∈ U (p) satisfying
f 00 (z)
2p
2pz
− 1+z 0
+
−
≺ φ(z).
f (z)
z − p 1 − pz
P∗
The
(p, w0 , φ) and
P bounds for w0 and some initial coefficients of f in
(p, φ) are obtained.
This research is supported by Short Term grant from Universiti Sains
Malaysia and also a grant from University of Delhi.
Coefficient Bounds
S. K. Lee, V. Ravichandran
and S. Shamani
vol. 10, iss. 3, art. 71, 2009
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Contents
1
Introduction
4
2
Coefficients Bound Problem
7
Coefficient Bounds
S. K. Lee, V. Ravichandran
and S. Shamani
vol. 10, iss. 3, art. 71, 2009
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1.
Introduction
Let U (p) denote the class of univalent meromorphic functions f in the unit disk ∆
with a simple pole at z = p > 0 and with the normalization f (0) = 0 and f 0 (0) = 1.
Let U ∗ (p, w0 ) be the subclass of U (p) such that f (z) ∈ U ∗ (p, w0 ) if and only if there
is a ρ, 0 < ρ < 1, with the property that
<
zf 0 (z)
<0
f (z) − w0
for ρ < |z| < 1. The functions in U ∗ (p, w0 ) map |z| < r < ρ (for some ρ, p <
ρ < 1) onto the complement of a set which is starlike with respect to w0 . Further
the functions in U ∗ (p, w0 ) all omit the value w0 . This class of starlike meromorphic
functions is developed from Robertson’s concept of star center points [11]. Let P
denote the class of functions P (z) which are meromorphic in ∆ and satisfy P (0) = 1
and <{P (z)} ≥ 0 for all z ∈ ∆.
For f (z) ∈ U ∗ (p, w0 ), there is a function P (z) ∈ P such that
f 0 (z)
p
pz
(1.1)
z
+
−
= −P (z)
f (z) − w0 z − p 1 − pz
P
for all z ∈ ∆. Let ∗ (p, w0 ) denote the class of functions f (z) which satisfy
(1.1)
P
and the condition f (0) = 0, f 0 (0) P
= 1. Then U ∗ (p, w0 ) is√a subset of ∗ (p, w0 ).
Miller [9] proved that U ∗ (p, w0 ) = ∗ (p, w0 ) for p ≤ 2 − 3.
Let K(p) denote the class of functions which belong to U (p) and map |z| < r < ρ
(for some p < ρ < 1) onto the complement of a convex set. If f ∈ K(p), then there
is a p < ρ < 1, such that for each z, ρ < |z| < 1
zf 00 (z)
< 1+ 0
≤ 0.
f (z)
Coefficient Bounds
S. K. Lee, V. Ravichandran
and S. Shamani
vol. 10, iss. 3, art. 71, 2009
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If f ∈ K(p), then for each z in ∆,
f 00 (z)
2p
2pz
(1.2)
< 1+z 0
+
−
≤ 0.
f (z)
z − p 1 − pz
P
Let (p) denote the class of functions f which satisfy
P (1.2) and the conditions
0
f (0) = 0 and f (0) = √
1. The classP
K(p) is contained in (p). Royster [12] showed
that for 0 < p ≤ 2 − P3, if f ∈ (p) and is meromorphic, then f ∈ K(p). Also,
for each function f ∈ (p), there is a function P ∈ P such that
1+z
f 00 (z)
2p
2pz
+
−
= −P (z).
f 0 (z)
z − p 1 − pz
The class U (p) and related classes have been studied in [3], [4], [5] and [6].
Let A be the class of all analytic functions of the form f (z) = z + a2 z 2 +
a3 z 3 + · · · in ∆. Several subclasses of univalent functions are characterized by the
quantities zf 0 (z)/f (z) or 1 + zf 00 (z)/f 0 (z) lying often in a region in the right-half
plane. Ma and Minda [7] gave a unified presentation of various subclasses of convex
and starlike functions. For an analytic function φ with positive real part on ∆ with
φ(0) = 1, φ0 (0) > 0 which maps the unit disk ∆ onto a region starlike (univalent)
with respect to 1 which is symmetric with respect to the real axis, they considered the
class S ∗ (φ) consisting of functions f ∈ A for which zf 0 (z)/f (z) ≺ φ(z) (z ∈ ∆).
They also investigated a corresponding class C(φ) of functions f ∈ A satisfying
1 + zf 00 (z)/f 0 (z) ≺ φ(z) (z ∈ ∆). For related results, see [1, 2, 8, 13]. In
the following definition, we consider the corresponding extension for meromorphic
univalent functions.
Definition 1.1. Let φ be a function with positive real part on ∆ with φ(0) = 1,
φ0 (0) > 0 which maps ∆ onto a region starlike with respect to 1 which is symmetric
Coefficient Bounds
S. K. Lee, V. Ravichandran
and S. Shamani
vol. 10, iss. 3, art. 71, 2009
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P
with respect to the real axis. The class ∗ (p, w0 , φ) consists of functions f ∈ U (p)
satisfying
zf 0 (z)
p
pz
−
+
−
≺ φ(z) (z ∈ ∆).
f (z) − w0 z − p 1 − pz
P
The class (p, φ) consists of functions f ∈ U (p) satisfying
f 00 (z)
2p
2pz
− 1+z 0
+
−
≺ φ(z) (z ∈ ∆).
f (z)
z − p 1 − pz
In this paper, thePbounds on |w0 | will
P be determined. Also the bounds for some
∗
coefficients of f in
(p, w0 , φ) and (p, φ) will be obtained.
Coefficient Bounds
S. K. Lee, V. Ravichandran
and S. Shamani
vol. 10, iss. 3, art. 71, 2009
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2.
Coefficients Bound Problem
To prove our main result, we need the following:
Lemma 2.1 ([7]). If p1 (z) = 1 + c1 z + c2 z 2 + · · · is a function with positive real
part in ∆, then

−4v + 2 if v ≤ 0,



2
if 0 ≤ v ≤ 1,
|c2 − vc21 | ≤



4v − 2
if v ≥ 1.
When v < 0 or v > 1, equality holds if and only if p1 (z) is (1 + z)/(1 − z) or one of
its rotations. If 0 < v < 1, then equality holds if and only if p1 (z) is (1+z 2 )/(1−z 2 )
or one of its rotations. If v = 0, the equality holds if and only if
1 1
1+z
1 1
1−z
p1 (z) =
+ λ
+
− λ
(0 ≤ λ ≤ 1)
2 2
1−z
2 2
1+z
or one of its rotations. If v = 1, the equality holds if and only if p1 is the reciprocal
of one of the functions such that equality holds in the case of v = 0.
Theorem 2.2. P
Let φ(z) = 1 + B1 z + B2 z 2 + · · · and f (z) = z + a2 z 2 + · · · in
|z| < p. If f ∈ ∗ (p, w0 , φ), then
2p
w0 =
pB1 c1 − 2p2 − 2
and
(2.1)
p2
p
p
≤ |w0 | ≤ 2
.
+ B1 p + 1
p − B1 p + 1
Coefficient Bounds
S. K. Lee, V. Ravichandran
and S. Shamani
vol. 10, iss. 3, art. 71, 2009
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Also, we have
(2.2)

 |w0 ||B2 |
2
a2 + w0 p2 + 1 + 1 ≤
2 2
 |w0 |B1
2
p
w0
2
if |B2 | ≥ B1 ,
if |B2 | ≤ B1 .
Proof. Let h be defined by
zf 0 (z)
p
pz
h(z) = −
+
−
= 1 + b1 z + b2 z 2 + · · · .
f (z) − w0 z − p 1 − pz
Coefficient Bounds
S. K. Lee, V. Ravichandran
and S. Shamani
vol. 10, iss. 3, art. 71, 2009
Then it follows that
1
1
+
, and
p w0
1
1
2a2
b2 = p2 + 2 + 2 +
.
p
w0
w0
b1 = p +
(2.3)
(2.4)
Since φ is univalent and h ≺ φ, the function
p1 (z) =
1 + φ−1 (h(z))
= 1 + c1 z + c2 z 2 + · · ·
1 − φ−1 (h(z))
is analytic and has a positive real part in ∆. Also, we have
p1 (z) − 1
(2.5)
h(z) = φ
p1 (z) + 1
and from this equation (2.5), we obtain
(2.6)
1
b1 = B1 c1
2
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and
1
1 2
1
b2 = B1 c2 − c1 + B2 c21 .
2
2
4
(2.7)
From (2.3), (2.4), (2.6) and (2.7), we get
w0 =
(2.8)
2p
pB1 c1 − 2p2 − 2
and
Coefficient Bounds
S. K. Lee, V. Ravichandran
and S. Shamani
vol. 10, iss. 3, art. 71, 2009
2
(2.9)
a2 =
p w0
w0
1
w0
(2B1 c2 − B1 c21 + B2 c21 ) −
− 2−
.
8
2
2p
2w0
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From (2.3) and (2.6), we obtain
p+
Contents
1
1
1
+
= B1 c1
p w0
2
and, since |c1 | ≤ 2 for a function with positive real part, we have
1
1
1
1
1
≤ p + +
≤ B1 |c1 | ≤ B1
p + −
p |w0 |
p w0 2
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or
−B1 ≤ p +
1
1
−
≤ B1 .
p |w0 |
Rewriting the inequality, we obtain
p2
p
p
≤ |w0 | ≤ 2
.
+ B1 p + 1
p − B1 p + 1
Close
From (2.9), we obtain
a2 + w0 p2 + 1 + 1 = w0 1 B1 c2 − 1 c21 + 1 B2 c21 2
p2 w02 2 2
2
4
|w0 |B1 B1 − B2 2 =
c2 −
c1 .
4
2B1
The result now follows from Lemma 2.1.
P
P
The classes ∗ (p, w0 , φ) and (p, φ) are indeed a more general class of functions, as can be seen in the following corollaries.
P
Corollary 2.3 ([10, inequality 4, p. 447]). If f (z) ∈ ∗ (p, w0 ), then
p
p
≤ |w0 | ≤
.
2
(1 + p)
(1 − p)2
Coefficient Bounds
S. K. Lee, V. Ravichandran
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vol. 10, iss. 3, art. 71, 2009
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Proof. Let B1 = 2 in (2.1) of Theorem 2.2.
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Corollary 2.4 ([10, Theorem 1, p. 447]). Let f ∈ ∗ (p, w0 ) and f (z) = z +a2 z 2 +
· · · in |z| < p. Then the second coefficient a2 is given by
1
1
1
2
a2 = w0 b2 − p − 2 − 2 ,
2
p
w0
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where the region of variability for a2 is contained in the disk
1
1
1
2
≤ |w0 |.
a2 + w0 p +
+
2
p2 w02 Proof. Let B1 = 2 in (2.2) of Theorem 2.2.
The next theorem is for convex meromorphic functions.
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Theorem 2.5. P
Let φ(z) = 1 + B1 z + B2 z 2 + · · · and f (z) = z + a2 z 2 + · · · in
|z| < p. If f ∈ (p, φ), then
2p2 − B1 p + 2
2p2 + B1 p + 2
≤ |a2 | ≤
.
2p
2p
Also
2 1
2 2
1 1 2
p + 2 − a2 − µ a2 − p −
a3 −
3
p
3
p  |2B +3µB 2 |
 2 12 1
≤
 B1
6
Coefficient Bounds
S. K. Lee, V. Ravichandran
and S. Shamani
vol. 10, iss. 3, art. 71, 2009
2
+ 3µB1 | ≥ 2,
if | 2B
B1
2
+ 3µB1 | ≤ 2.
if | 2B
B1
Proof. Let h now be defined by
2p
2pz
zf 00 (z)
+
−
= 1 + b1 z + b2 z 2 + · · ·
h(z) = − 1 + 0
f (z)
z − p 1 − pz
and p1 be defined as in the proof of Theorem 2.2. A computation shows that
1
(2.10)
b1 = 2 p + − a2 , and
p
1
2
2
(2.11)
b2 = 2 p + 2 + 2a2 − 3a3 .
p
From (2.6) and (2.10), we have
(2.12)
a2 = p +
1 B1 c1
−
.
p
4
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From (2.7) and (2.11), we have
8
1
2
2
2
2
(2.13)
a3 =
8p + 2 + 16a2 − 2B1 c2 + B1 c1 − B2 c1 .
24
p
From (2.12), we have
2p +
or
2
1
− 2a2 = B1 c1
p
2
2p + 2 − 2|a2 | ≤ |2p + 2 − 2a2 | ≤ 1 B1 |c1 | ≤ B1 .
p
p
2
Coefficient Bounds
S. K. Lee, V. Ravichandran
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vol. 10, iss. 3, art. 71, 2009
Thus we have
−B1 ≤ 2p + (2/p) − 2|a2 | ≤ B1
or
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2
2
2p + B1 p + 2
2p − B1 p + 2
≤ |a2 | ≤
.
2p
2p
From (2.12) and (2.13), we obtain
2 1 2
1
2
1 p + 2 − a22 − µ a2 − p −
a3 −
3
p
3
p 2 2 1
B1 c1 2
2
−2B1 c2 + B1 c1 − B2 c1 − µ
=
24
16 B1 1
B2
3µB1 2 =
c
−
−
−
c1 .
2
12 2 2B1
4
The result now follows from Lemma 2.1.
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References
[1] R.M. ALI, V. RAVICHANDRAN AND N. SEENIVASAGAN, Coefficient
bounds for p-valent functions, Appl. Math. Comput., 187(1) (2007), 35–46.
[2] R.M. ALI, V. RAVICHANDRAN, AND S.K. LEE, Subclasses of multivalent starlike and convex functions, Bull. Belgian Math. Soc. Simon Stevin, 16
(2009), 385–394.
Coefficient Bounds
S. K. Lee, V. Ravichandran
and S. Shamani
[3] A.W. GOODMAN, Functions typically-real and meromorphic in the unit circle,
Trans. Amer. Math. Soc., 81 (1956), 92–105.
vol. 10, iss. 3, art. 71, 2009
[4] J.A. JENKINS, On a conjecture of Goodman concerning meromorphic univalent functions, Michigan Math. J., 9 (1962), 25–27.
Title Page
[5] Y. KOMATU, Note on the theory of conformal representation by meromorphic
functions. I, Proc. Japan Acad., 21 (1945), 269–277.
[6] K. LADEGAST, Beiträge zur Theorie der schlichten Funktionen, Math. Z., 58
(1953), 115–159.
[7] W. MA AND D. MINDA, A unified treatment of some special classes of univalent functions, in: Proceedings of the Conference on Complex Analysis, Z. Li,
F. Ren, L. Yang, and S. Zhang (Eds.), Int. Press (1994), 157–169.
[8] M.H. MOHD, R.M. ALI, S.K. LEE AND V. RAVICHANDRAN, Subclasses
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Math. Soc., 25 (1970), 220–228.
[10] J. MILLER, Starlike meromorphic functions, Proc. Amer. Math. Soc., 31
(1972), 446–452.
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[11] M.S. ROBERTSON, Star center points of multivalent functions, Duke Math. J.,
12 (1945), 669–684.
[12] W.C. ROYSTER, Convex meromorphic functions, in Mathematical Essays
Dedicated to A. J. Macintyre, 331–339, Ohio Univ. Press, Athens, Ohio (1970).
[13] S. SHAMANI, R.M. ALI, S.K. LEE AND V. RAVICHANDRAN, Convolution
and differential subordination for multivalent functions, Bull. Malays. Math.
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Coefficient Bounds
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and S. Shamani
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