Volume 10 (2009), Issue 3, Article 71, 6 pp.
COEFFICIENT BOUNDS FOR MEROMORPHIC STARLIKE AND CONVEX
FUNCTIONS
SEE KEONG LEE, V. RAVICHANDRAN, AND SUPRAMANIAM SHAMANI
U NIVERSITI S AINS M ALAYSIA
11800 USM P ENANG , M ALAYSIA
sklee@cs.usm.my
D EPARTMENT OF M ATHEMATICS
U NIVERSITY OF D ELHI
D ELHI 110 007, I NDIA
vravi@maths.du.ac.in
URL: http://people.du.ac.in/~ vravi
S CHOOL OF M ATHEMATICAL S CIENCES
U NIVERSITI S AINS M ALAYSIA
11800 USM P ENANG , M ALAYSIA
sham105@hotmail.com
Received 30 January, 2008; accepted 03 May, 2009
Communicated by S.S. Dragomir
A BSTRACT. 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
P∗a region starlike with respect to 1 which is symmetric
with respect to the real axis. The class
(p, w0 , φ) consists of functions f ∈ U (p) satisfying
0
zf (z)
p
pz
−
+
−
≺ 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∗
P
The bounds for w0 and some initial coefficients of f in
(p, w0 , φ) and (p, φ) are obtained.
Key words and phrases: Univalent meromorphic functions; starlike function, convex function, Fekete-Szegö inequality.
2000 Mathematics Subject Classification. Primary 30C45, Secondary 30C80.
This research is supported by Short Term grant from Universiti Sains Malaysia and also a grant from University of Delhi.
034-08
2
S. K. L EE , V. R AVICHANDRAN , AND S. S HAMANI
1. I NTRODUCTION
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
(1.1)
z
p
pz
f 0 (z)
+
−
= −P (z)
f (z) − w0 z − p 1 − pz
P
for all z ∈ ∆. Let ∗ (p, w0 ) denote the class of functions P
f (z) which satisfy (1.1) and the
0
∗
condition f (0)
= 0, f (0) = 1. Then √
U (p, w0 ) is a subset of ∗ (p, w0 ). Miller [9] proved that
P
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)
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 P
which satisfy (1.2) and the conditions f (0) = 0 √
and
0
f (0) =P1. The class K(p) is contained in (p). Royster [12] showed that for 0P
< p ≤ 2 − 3,
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).
0
f (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.
J. Inequal. Pure and Appl. Math., 10(3) (2009), Art. 71, 6 pp.
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C OEFFICIENT B OUNDS FOR M EROMORPHIC S TARLIKE AND C ONVEX F UNCTIONS
3
Definition 1.1. Let φ be a function with positive real part on ∆ with φ(0) = 1, φ0 (0) > 0 which
maps ∆ onto
P a region starlike with 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
zf 0 (z)
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
+
−
≺ φ(z) (z ∈ ∆).
− 1+z 0
f (z)
z − p 1 − pz
In this
the bounds
Ppaper,
Pon |w0 | will be determined. Also the bounds for some coefficients
∗
of f in
(p, w0 , φ) and (p, φ) will be obtained.
2. C OEFFICIENTS B OUND P ROBLEM
To prove our main result, we need the following:
Lemma 2.1 ([7]). If p1 (z) = 1 + c1 z + c2 z 2 + · · ·
then

 −4v + 2


2
2
|c2 − vc1 | ≤



4v − 2
is a function with positive real part in ∆,
if
v ≤ 0,
if
0 ≤ v ≤ 1,
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+z
1 1
1−z
1 1
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.
2
2
Theorem
P∗ 2.2. Let φ(z) = 1 + B1 z + B2 z + · · · and f (z) = z + a2 z + · · · in |z| < p. If
f∈
(p, w0 , φ), then
2p
w0 =
pB1 c1 − 2p2 − 2
and
p
p
(2.1)
≤ |w0 | ≤ 2
.
2
p + B1 p + 1
p − B1 p + 1
Also, we have
(2.2)

 |w0 ||B2 |
2
a2 + w0 p2 + 1 + 1 ≤
2
p2 w02  |w0 |B1
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
J. Inequal. Pure and Appl. Math., 10(3) (2009), Art. 71, 6 pp.
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4
S. K. L EE , V. R AVICHANDRAN , AND S. S HAMANI
Then it follows that
1
1
+
, and
p w0
1
2a2
1
b2 = p 2 + 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
1
b1 = B1 c1
2
(2.6)
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
w0
p2 w0
w0
1
(2B1 c2 − B1 c21 + B2 c21 ) −
− 2−
.
8
2
2p
2w0
From (2.3) and (2.6), we obtain
1
1
1
p+ +
= B1 c1
p w0
2
(2.9)
a2 =
and, since |c1 | ≤ 2 for a function with positive real part, we have
1
1
1
1
1
p + −
≤ p + +
≤ B1 |c1 | ≤ B1
p |w0 |
p w0 2
or
−B1 ≤ p +
1
1
−
≤ B1 .
p |w0 |
Rewriting the inequality, we obtain
p
p
≤ |w0 | ≤ 2
.
2
p + B1 p + 1
p − B1 p + 1
From (2.9), we obtain
w0 1
w
1
1
1 2
1
0
2
2 a2 +
p
+
+
=
B
c
−
c
+
B
c
1
2
2
1
1
2
p2 w02 2 2
2
4
|w0 |B1 B1 − B2 2 =
c2 −
c1 .
4
2B1
The result now follows from Lemma 2.1.
J. Inequal. Pure and Appl. Math., 10(3) (2009), Art. 71, 6 pp.
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C OEFFICIENT B OUNDS FOR M EROMORPHIC S TARLIKE AND C ONVEX F UNCTIONS
5
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
Proof. Let B1 = 2 in (2.1) of Theorem 2.2.
P∗
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
where the region of variability for a2 is contained in the disk
1
1
1
2
≤ |w0 |.
a2 + w0 p +
+
2
p2 w 2 0
Proof. Let B1 = 2 in (2.2) of Theorem 2.2.
The next theorem is for convex meromorphic functions.
2
2
Theorem
P 2.5. Let φ(z) = 1 + B1 z + B2 z + · · · and f (z) = z + a2 z + · · · in |z| < p. If
f ∈ (p, φ), then
2p2 − B1 p + 2
2p2 + B1 p + 2
≤ |a2 | ≤
.
2p
2p
Also

2  |2B2 +3µB12 | if | 2B2 + 3µB | ≥ 2,
1
12
B1
1
2
1 1 2
p + 2 − a22 − µ a2 − p −
≤
a3 −
3
p
3
p  B1
2
if | 2B
+ 3µB1 | ≤ 2.
6
B1
Proof. Let h now be defined by
zf 00 (z)
2p
2pz
h(z) = − 1 + 0
+
−
= 1 + b1 z + b2 z 2 + · · ·
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
From (2.7) and (2.11), we have
1
8
2
2
2
2
8p + 2 + 16a2 − 2B1 c2 + B1 c1 − B2 c1 .
(2.13)
a3 =
24
p
From (2.12), we have
2p +
J. Inequal. Pure and Appl. Math., 10(3) (2009), Art. 71, 6 pp.
2
1
− 2a2 = B1 c1
p
2
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6
S. K. L EE , V. R AVICHANDRAN , AND S. S HAMANI
or
2
2p + − 2|a2 | ≤ |2p + 2 − 2a2 | ≤ 1 B1 |c1 | ≤ B1 .
p
p
2
Thus we have
−B1 ≤ 2p + (2/p) − 2|a2 | ≤ B1
or
2p2 + B1 p + 2
2p2 − B1 p + 2
≤ |a2 | ≤
.
2p
2p
From (2.12) and (2.13), we obtain
2 1
2
1
1
p2 + 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.
R EFERENCES
[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.
[3] A.W. GOODMAN, Functions typically-real and meromorphic in the unit circle, Trans. Amer. Math.
Soc., 81 (1956), 92–105.
[4] J.A. JENKINS, On a conjecture of Goodman concerning meromorphic univalent functions, Michigan Math. J., 9 (1962), 25–27.
[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 of meromorphic functions associated with convolution, J. Inequal. Appl., 2009 (2009), Article ID 190291, 10 pp.
[9] J. MILLER, Convex meromorphic mappings and related functions, Proc. Amer. Math. Soc., 25
(1970), 220–228.
[10] J. MILLER, Starlike meromorphic functions, Proc. Amer. Math. Soc., 31 (1972), 446–452.
[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. Sci. Soc. (2), 32(3) (2009), to appear.
J. Inequal. Pure and Appl. Math., 10(3) (2009), Art. 71, 6 pp.
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