Volume 2008, Article ID 931981,9pages doi:10.1155/2008/931981
Research Article
Coefficient Bounds for Certain Classes of
Meromorphic Functions
H. Silverman,1 K. Suchithra,2B. Adolf Stephen,3 and A. Gangadharan2
1Department of Mathematics, College of Charleston, Charleston, SC 29424, USA
2Department of Applied Mathematics, Sri Venkateswara College of Engineering, Sriperumbudur,
Chennai 602105, Tamilnadu, India
3School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Penang, Malaysia
Correspondence should be addressed to H. Silverman,silvermanh@cofc.edu Received 19 May 2008; Revised 9 September 2008; Accepted 4 December 2008
Recommended by Ramm Mohapatra
Sharp bounds for |a1 − μa20| are derived for certain classes Σ∗φ and Σ∗αφ of meromor-phic functions fz defined on the punctured open unit disk for which −zfz/fz and
−1−2αzfz αz2fz/1−αfz−αzfz α∈ C−0,1;Rα ≥ 0, respectively, lie
in a region starlike with respect to 1 and symmetric with respect to the real axis. Also, certain applications of the main results for a class of functions defined through Ruscheweyh derivatives are obtained.
Copyrightq2008 H. Silverman et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
LetΣdenote the class of functions of the form
fz 1z∞
k0
akzk, 1.1
which areanalyticandunivalentin the punctured open unit disk
Δ∗z∈C: 0<|z|<1 Δ− {0}, 1.2
whereΔis the open unit diskΔ {z∈C:|z|<1}.
A functionf ∈Σis said to bemeromorphic univalent starlike of orderαif
and the class of all such meromorphic univalent starlike functions in Δ∗ is denoted by Σ∗α.
Recently, Uralegaddi and Desai 1 studied the class Σα, β of functions f ∈ Σ satisfying the condition
zfz/fz 1
zfz/fz 2α−1
≤β z∈Δ; 0≤α <1; 0< β≤1. 1.4
Kulkarni and Joshi 2studied the classΣα, β, γof functionsf∈Σsatisfying the condition
zfz/fz 1
2γzfz/fz α−zfz/fz 1 ≤β
z∈Δ; 0≤α <1; 0< β≤1; 1
2 < γ≤1 . 1.5
Earlier, several authors 3–6have studied similar subclasses ofΣ∗α.
LetS consist of functionsfz z∞k2akzk which are analytic and univalent in
Δ. Many researchers including 7–11 have obtained Fekete-Szeg ¨o inequality for analytic functionsf ∈ S.
In this paper, we obtain Fekete-Szeg ¨o-like inequalities for new classes of meromorphic functions, which are defined in what follows. Also, we give applications of our results to certain functions defined through Ruscheweyh derivatives.
Definition 1.1. Letφzbe an analytic function with positive real part on Δwith φ0 1, φ0>0, which maps the unit diskΔonto a region starlike with respect to 1, and is symmetric
with respect to the real axis. LetΣ∗φbe the class of functionsf ∈Σfor which
−zfz
fz ≺φz z∈Δ, 1.6
where≺denotes subordination between analytic functions.
The above-defined class Σ∗φ is the meromorphic analogue of the class S∗φ, introduced and studied by Ma and Minda 8, which consists of functionsf ∈ Sfor which zfz/fz≺φz, z∈Δ.
More generally, under the same conditions asDefinition 1.1, we add a parameter.
Definition 1.2. LetΣ∗αφbe the class of functionsf∈Σfor which
−1−2αzfz αz2fz
1−αfz−αzfz ≺φz
Some of the interesting subclasses ofΣ∗αφare
1 Σ∗
0φ Σ∗φ,
2 Σ∗
01 1−2αz/1−z Σ∗α, 0≤α <1,
3 Σ∗
01β1−2αγz/1β1−2γz Σα, β, γ, 0≤α <1,0< β≤1,1/2≤γ≤1
studied by Kulkarni and Joshi 2,
4 Σ∗
01Awz/1Bwz K1A, B, 0 ≤ B < 1; −B < A < Bstudied by
Karunakaran 12.
To prove our result, we need the following lemma.
Lemma 1.3see 13. Ifpz 1c1zc2z2c3z3· · · is a function with positive real part in
Δ, then for any complex numberμ,
c2−μc21≤2 max
1,|1−2μ|. 1.8
2. Coefficient bounds
By making use ofLemma 1.3, we prove the following bounds for the classesΣ∗φandΣ∗αφ.
Theorem 2.1. Letφz 1B1zB2z2· · ·. Iffzgiven by1.1belongs toΣ∗φ, then for any complex numberμ,
i a1−μa20≤ B1
2 max
1,B2
B1 −1−2μB1
, B1/0, 2.1
ii a1−μa20≤1, B10. 2.2
The bounds are sharp.
Proof. Iffz∈Σ∗φ, then there is a Schwarz functionwz, analytic inΔwithw0 0 and |wz|<1 inΔsuch that
−zffzz φwz. 2.3
Define the functionpzby
pz 1wz
1−wz 1c1zc2z
Sincewzis a Schwarz function, we see thatRpz>0 andp0 1. Therefore,
φwzφ
pz−1
pz 1
φ
1 2
c1z
c2−
c2 1 2 z2 c3 c3 1
4 −c1c2
z3· · ·
11
2B1c1z
1 2B1
c2−
1 2c
2
1
1 4B2c
2
1 z2· · ·.
2.5
Now by substituting2.5in2.3, we have
−zffzz 1 1
2B1c1z
1 2B1
c2−
1 2c
2
1
1 4B2c
2
1 z2· · ·. 2.6
From this equation and1.1, we obtain
a0
B1c1
2 0,
−a1a1
a0B1c1
2 B1c2
2 − B1c21
4 B2c21
4 .
2.7
Or equivalently,
a0−
1 2B1c1,
a1−1
2
1 2B1c2
1 4
B2−B1−B12 c2 1 . 2.8 Therefore,
a1−μa20−
B1
4
c2−vc21
, 2.9
where
v 1 2
1−B2
B1 1−2μB1
. 2.10
Now, the result 2.1follows by an application of Lemma 1.3. Also, if B1 0, then
a00 anda1 −1/8B2c21.
Sincepzhas positive real part,|c1| ≤2, so that|a1−μa20| ≤ |B2|/2.Sinceφzalso has
The bounds are sharp for the functionsF1zandF2zdefined by
−zF1z
F1z
φz2, whereF 1z
1z2
z1−z2,
−zF2z
F2z φz,
whereF2z 1
z z1−z.
2.11
Clearly, the functionsF1z, F2z∈Σ.
Proceeding similarly, we now obtain the bounds for the classΣ∗αφ.
Theorem 2.2. Letφz 1B1zB2z2· · ·. Iffzgiven by1.1belongs toΣ∗αφ, then for any complex numberμ,
i a1−μa20≤
B1
21−2α
max
1,B2
B1 −
1−21−2α 1−α2 μ
B1
, B1/0, 2.12
ii a1−μa20≤
1−12α, B10. 2.13
The bounds obtained are sharp.
Proof. Iffz∈Σ∗αφ, then there is a Schwarz functionwz, analytic inΔwithw0 0 and
|wz|<1 inΔsuch that
−1−2αzfz αz2fz
1−αfz−αzfz φ
wz, α∈C−0,1,Rα≥0. 2.14
Now using2.5and1.1in2.14, and comparing the coefficients, we have
a01−α 1
2B1c1 0,
−a11−2α a11−2α 1
2a01−αB1c1 1 2B1c2−
1 4
B1−B2
c2 1;
2.15
or equivalently,
a0−211−αB1c1,
a1−
1 21−2α
1 2B1c2
1 4
B2−B1−B12
c2 1 .
2.16
Therefore,
a1−μa20 −
B1
41−2α
c2−vc12
where
v 1 2
1−B2 B1
1− 21−2α 1−α2 μ
B1
. 2.18
Now, the result2.12follows by an application ofLemma 1.3. Also, ifB10, thena00 and
a1 −1/81−2αB2c21.
Sincepzhas positive real part,|c1| ≤2, so that|a1−μa20| ≤ |B2|/21−2α.Sinceφz
also has positive real part,|B2| ≤2. Thus,|a1−μa20| ≤ |1/1−2α|, proving2.13.
The bounds are sharp for the functionsF1zandF2zdefined by
−1−2αzF
1z αz2F1z
1−αF1z−αzF1z
φz2, whereF
1z 1z
2
z1−z2,
−1−2αzF
2z αz2F2z
1−αF2z−αzF2z φz,
whereF2z 1
z z1−z.
2.19
ClearlyF1z, F2z∈Σ.
Remark 2.3. By puttingα0 in2.12and2.13, we get the results2.1and2.2.
3. Applications to functions defined by Ruscheweyh derivatives
In this section, we introduce two classesΣ∗λφandΣ∗α,λφof meromorphic functions defined by Ruscheweyh derivatives, and obtain coefficient bounds for functions in these classes.
Letf∈Σbe given by2.1andg ∈Σbe given by
gz 1z∞
k0
bkzk, 3.1
then the Hadamard product offandgis defined as
f∗gz 1z∞
k0
akbkzk g∗fz. 3.2
In terms of the Hadamard product of two functions, the analogue of the familiar Ruscheweyh derivative 14is defined as
Dλfz: 1
so that
Dλfz 1
z
zλ1fz
λ!
λ
λ >−1;f ∈Σ, 3.4
where, here and in what followsλis an integer>−1, that is,λ∈N0{0,1,2, . . .}.
It follows from3.3and3.4that
Dλfz 1
z
∞
k0
δλ, kakzk f∈Σ, 3.5
wheref∈Σis given by1.1and
δλ, k:
λk
1
k1 . 3.6
The above-defined operator Dλ for λ ∈ N0 {0,1,2, . . .} was also studied by Cho 15
and Padmanabhan 16. For various developments involving the operatorDλfor functions belonging toΣ, the reader may be referred to the recent works of Uralegaddi et al. 17–19
and others 20–22.
Using3.5, under the same conditions asDefinition 1.1, we define the classesΣ∗λφ andΣ∗α,λφas follows.
Definition 3.1. A functionf ∈Σis in the classΣ∗λφif
−z
Dλfz
Dλfz ≺φz z∈Δ. 3.7
Definition 3.2. A functionf ∈Σis in the classΣ∗α,λφif
−1−2αzDλfzαz2Dλfz
1−αDλfz−αzDλfz ≺φz,
z∈Δ;α∈C−0,1; Rα≥0. 3.8
For the classes Σ∗λφ and Σ∗α,λφ, using methods similar to those in the proof of
Theorem 2.1, we obtain the following results.
Theorem 3.3. Letφz 1B1zB2z2· · ·. Iffzgiven by1.1belongs toΣ∗λφ, then for any complex numberμ,
i a1−μa20≤
B1
λ1λ2
max
1,B2
B1 −
1−
λ
2 λ1 μ B1
, B1/0, 3.9
ii a1−μa20≤
λ1λ2 2, B10. 3.10
Theorem 3.4. Letφz 1B1zB2z2· · ·. Iffzgiven by1.1belongs toΣ∗α,λφ, then for any complex numberμ,
i a1−μa20≤
B1
1−2αλ1λ2
×max
1,B2
B1 −
1−1−2α 1−α2
λ
2 λ1 μ
B1
, B1/0,
3.11
ii a1−μa20≤
1−2αλ21λ2, B10. 3.12
The bounds are sharp.
Remark 3.5. Forλ0 in3.9,3.11, we get the results2.1and2.12, respectively. Also, for
αλ0 in3.11, we get the result2.1.
Acknowledgment
The authors are grateful to the referees for their useful comments.
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