COEFFICIENT BOUNDS FOR MEROMORPHIC STARLIKE AND CONVEX FUNCTIONS
SEE KEONG LEE, V. RAVICHANDRAN, AND SUPRAMANIAM SHAMANI
UNIVERSITISAINSMALAYSIA
11800 USM PENANG, MALAYSIA
sklee@cs.usm.my
DEPARTMENT OFMATHEMATICS
UNIVERSITY OFDELHI
DELHI110 007, INDIA
vravi@maths.du.ac.in
URL:http://people.du.ac.in/ vravi
SCHOOL OFMATHEMATICALSCIENCES
UNIVERSITISAINSMALAYSIA
11800 USM PENANG, MALAYSIA
sham105@hotmail.com
Received 30 January, 2008; accepted 03 May, 2009 Communicated by S.S. Dragomir
ABSTRACT. In this paper, some subclasses of meromorphic univalent functions in the unit disk
∆are extended. LetU(p)denote the class of normalized univalent meromorphic functionsf in∆with a simple pole atz = p > 0. Letφbe a function with positive real part on∆with φ(0) = 1,φ0(0)>0which maps∆onto a region starlike with respect to1which is symmetric with respect to the real axis. The classP∗
(p, w0, φ)consists of functionsf ∈U(p)satisfying
−
zf0(z) f(z)−w0
+ p
z−p− pz 1−pz
≺φ(z).
The classP
(p, φ)consists of functionsf ∈U(p)satisfying
−
1 +zf00(z) f0(z) + 2p
z−p− 2pz 1−pz
≺φ(z).
The bounds forw0and some initial coefficients off inP∗
(p, w0, φ)andP(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
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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 atz =p > 0and with the normalizationf(0) = 0andf0(0) = 1. LetU∗(p, w0)be the subclass ofU(p)such thatf(z)∈U∗(p, w0)if and only if there is aρ,0< ρ <1, with the property that
< zf0(z) f(z)−w0 <0
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 valuew0. This class of starlike meromorphic functions is developed from Robertson’s concept of star center points [11]. LetP denote the class of functionsP(z)which are meromorphic in∆and satisfyP(0) = 1and<{P(z)} ≥0for allz ∈∆.
Forf(z)∈U∗(p, w0), there is a functionP(z)∈ P such that
(1.1) z f0(z)
f(z)−w0 + p
z−p− pz
1−pz =−P(z) for all z ∈ ∆. Let P∗
(p, w0) denote the class of functions f(z) which satisfy (1.1) and the conditionf(0) = 0, f0(0) = 1. ThenU∗(p, w0)is a subset ofP∗
(p, w0). Miller [9] proved that U∗(p, w0) = P∗
(p, w0)forp≤2−√ 3.
LetK(p)denote the class of functions which belong toU(p)and map|z|< r < ρ(for some p < ρ <1) onto the complement of a convex set. Iff ∈K(p), then there is ap < ρ <1, such that for eachz,ρ <|z|<1
<
1 + zf00(z) f0(z)
≤0.
Iff ∈K(p),then for eachz in∆,
(1.2) <
1 +zf00(z)
f0(z) + 2p
z−p − 2pz 1−pz
≤0.
LetP
(p)denote the class of functionsf which satisfy (1.2) and the conditionsf(0) = 0and f0(0) = 1.The classK(p)is contained inP
(p).Royster [12] showed that for0< p≤2−√ 3, iff ∈P
(p)and is meromorphic, thenf ∈K(p).Also, for each functionf ∈P
(p),there is a functionP ∈ P such that
1 +zf00(z)
f0(z) + 2p
z−p − 2pz
1−pz =−P(z).
The classU(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 +a2z2 +a3z3 +· · · in
∆. Several subclasses of univalent functions are characterized by the quantities zf0(z)/f(z) or 1 +zf00(z)/f0(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 ana- lytic 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 zf0(z)/f(z)≺φ(z) (z ∈∆). They also investigated a corresponding classC(φ)of functions f ∈ Asatisfying1 +zf00(z)/f0(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. Letφbe a function with positive real part on∆withφ(0) = 1,φ0(0)>0which maps∆onto a region starlike with respect to1which is symmetric with respect to the real axis.
The classP∗
(p, w0, φ)consists of functionsf ∈U(p)satisfying
−
zf0(z)
f(z)−w0 + p
z−p − pz 1−pz
≺φ(z) (z ∈∆).
The classP
(p, φ)consists of functionsf ∈U(p)satisfying
−
1 +zf00(z)
f0(z) + 2p
z−p − 2pz 1−pz
≺φ(z) (z ∈∆).
In this paper, the bounds on|w0|will be determined. Also the bounds for some coefficients off inP∗
(p, w0, φ)andP
(p, φ)will be obtained.
2. COEFFICIENTS BOUNDPROBLEM
To prove our main result, we need the following:
Lemma 2.1 ([7]). Ifp1(z) = 1 +c1z +c2z2+· · · is a function with positive real part in∆, then
|c2−vc21| ≤
−4v+ 2 if v ≤0, 2 if 0≤v ≤1,
4v−2 if v ≥1.
Whenv <0orv >1, equality holds if and only ifp1(z)is(1+z)/(1−z)or one of its rotations.
If0< v <1, then equality holds if and only ifp1(z)is(1 +z2)/(1−z2)or one of its rotations.
Ifv = 0, the equality holds if and only if p1(z) =
1 2 +1
2λ
1 +z 1−z +
1 2 − 1
2λ
1−z
1 +z (0≤λ≤1)
or one of its rotations. Ifv = 1, the equality holds if and only ifp1 is the reciprocal of one of the functions such that equality holds in the case ofv = 0.
Theorem 2.2. Letφ(z) = 1 +B1z +B2z2 +· · · andf(z) = z +a2z2 +· · · in|z| < p. If f ∈P∗
(p, w0, φ), then
w0 = 2p
pB1c1−2p2−2 and
(2.1) p
p2+B1p+ 1 ≤ |w0| ≤ p
p2−B1p+ 1. Also, we have
(2.2)
a2+ w0 2
p2+ 1 p2 + 1
w02
≤
|w0||B2|
2 if |B2| ≥B1,
|w0|B1
2 if |B2| ≤B1. Proof. Lethbe defined by
h(z) =−
zf0(z)
f(z)−w0 + p
z−p − pz 1−pz
= 1 +b1z+b2z2+· · · .
Then it follows that
b1 =p+ 1 p+ 1
w0, and (2.3)
b2 =p2+ 1 p2 + 1
w02 +2a2 w0. (2.4)
Sinceφis univalent andh≺φ, the function p1(z) = 1 +φ−1(h(z))
1−φ−1(h(z)) = 1 +c1z+c2z2+· · · is analytic and has a positive real part in∆. Also, we have
(2.5) h(z) =φ
p1(z)−1 p1(z) + 1
and from this equation (2.5), we obtain
(2.6) b1 = 1
2B1c1 and
(2.7) b2 = 1
2B1
c2− 1 2c21
+ 1
4B2c21. From (2.3), (2.4), (2.6) and (2.7), we get
(2.8) w0 = 2p
pB1c1−2p2−2 and
(2.9) a2 = w0
8 (2B1c2−B1c21+B2c21)− p2w0 2 − w0
2p2 − 1 2w0
. From (2.3) and (2.6), we obtain
p+1 p + 1
w0 = 1 2B1c1
and, since|c1| ≤2for a function with positive real part, we have
p+ 1 p− 1
|w0|
≤
p+ 1 p+ 1
w0
≤ 1
2B1|c1| ≤B1 or
−B1 ≤p+1 p − 1
|w0| ≤B1. Rewriting the inequality, we obtain
p
p2+B1p+ 1 ≤ |w0| ≤ p
p2−B1p+ 1. From (2.9), we obtain
a2+ w0 2
p2+ 1 p2 + 1
w20
=
w0 2
1 2B1
c2−1
2c21
+1 4B2c21
= |w0|B1 4
c2−
B1−B2 2B1
c21
.
The result now follows from Lemma 2.1.
The classes P∗
(p, w0, φ)and P
(p, φ) are indeed a more general class of functions, as can be seen in the following corollaries.
Corollary 2.3 ([10, inequality 4, p. 447]). Iff(z)∈P∗
(p, w0), then p
(1 +p)2 ≤ |w0| ≤ p (1−p)2.
Proof. LetB1 = 2in (2.1) of Theorem 2.2.
Corollary 2.4 ([10, Theorem 1, p. 447]). Let f ∈ P∗
(p, w0)and f(z) = z +a2z2 +· · · in
|z|< p. Then the second coefficienta2is given by a2 = 1
2w0
b2 −p2− 1 p2 − 1
w02
,
where the region of variability fora2 is contained in the disk
a2+ 1 2w0
p2+ 1 p2 + 1
w02
≤ |w0|.
Proof. LetB1 = 2in (2.2) of Theorem 2.2.
The next theorem is for convex meromorphic functions.
Theorem 2.5. Letφ(z) = 1 +B1z +B2z2 +· · · andf(z) = z +a2z2 +· · · in|z| < p. If f ∈P
(p, φ), then
2p2−B1p+ 2
2p ≤ |a2| ≤ 2p2+B1p+ 2
2p .
Also
a3− 1 3
p2+ 1 p2
−2
3a22−µ
a2−p− 1 p
2
≤
|2B2+3µB12|
12 if |2BB2
1 + 3µB1| ≥2,
B1
6 if |2BB2
1 + 3µB1| ≤2.
Proof. Lethnow be defined by h(z) = −
1 + zf00(z)
f0(z) + 2p
z−p − 2pz 1−pz
= 1 +b1z+b2z2+· · ·
andp1 be defined as in the proof of Theorem 2.2. A computation shows that b1 = 2
p+1
p −a2
, and (2.10)
b2 = 2
p2+ 1
p2 + 2a22 −3a3 (2.11) .
From (2.6) and (2.10), we have
(2.12) a2 =p+1
p − B1c1
4 . From (2.7) and (2.11), we have
(2.13) a3 = 1
24
8p2 + 8
p2 + 16a22−2B1c2+B1c21−B2c21
. From (2.12), we have
2p+2
p −2a2 = 1 2B1c1
or
2p+2
p −2|a2|
≤ |2p+ 2
p−2a2| ≤ 1
2B1|c1| ≤B1. Thus we have
−B1 ≤2p+ (2/p)−2|a2| ≤B1 or
2p2−B1p+ 2
2p ≤ |a2| ≤ 2p2+B1p+ 2
2p .
From (2.12) and (2.13), we obtain
a3− 1 3
p2+ 1 p2
−2
3a22−µ
a2−p− 1 p
2
=
1
24 −2B1c2+B1c21−B2c21
−µ
B12c21 16
= B1 12
c2− 1
2− B2
2B1 − 3µB1 4
c21
.
The result now follows from Lemma 2.1.
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