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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/1αfzαzfz α C0,1;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∗zC: 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

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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 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

zfz/fz αzfz/fz 1β

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 zk2akzk 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/fzφz, zΔ.

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−αfzαzfzφz

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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

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Sincewzis a Schwarz function, we see thatRpz>0 andp0 1. Therefore,

φwzφ

pz1

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

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The bounds are sharp for the functionsF1zandF2zdefined by

zF1z

F1z

φz2, whereF 1z

1z2

z1−z2,

zF2z

F2z φz,

whereF2z 1

z z1z.

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≤

11, 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−αfzαzfz 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,

a0−211αB1c1,

a1−

1 21−2α

1 2B1c2

1 4

B2−B1−B12

c2 1 .

2.16

Therefore,

a1−μa20 −

B1

41−2α

c2−vc12

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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 z1z.

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

fz: 1

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so that

fz 1

z

1fz

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

fz 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 for λ ∈ N0 {0,1,2, . . .} was also studied by Cho 15

and Padmanabhan 16. For various developments involving the operatorfor 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

fz

fzφz z∈Δ. 3.7

Definition 3.2. A functionf ∈Σis in the classΣ∗α,λφif

−1−2αzfzαz2Dλfz

1−αDλfzαzDλfzφz,

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≤

λ2 2, B10. 3.10

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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≤

12αλ22, 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.

References

1 B. A. Uralegaddi and A. R. Desai, “Integrals of meromorphic starlike functions with positive and fixed second coefficients,”The Journal of the Indian Academy of Mathematics, vol. 24, no. 1, pp. 27–36, 2002.

2 S. R. Kulkarni and S. S. Joshi, “On a subclass of meromorphic univalent functions with positive coefficients,”The Journal of the Indian Academy of Mathematics, vol. 24, no. 1, pp. 197–205, 2002. 3 J. Clunie, “On meromorphic schlicht functions,”Journal of the London Mathematical Society, vol. s1-34,

no. 2, pp. 215–216, 1959.

4 J. Miller, “Convex meromorphic mappings and related functions,” Proceedings of the American Mathematical Society, vol. 25, no. 2, pp. 220–228, 1970.

5 Ch. Pommerenke, “On meromorphic starlike functions,”Pacific Journal of Mathematics, vol. 13, no. 1, pp. 221–235, 1963.

6 W. C. Royster, “Meromorphic starlike multivalent functions,”Transactions of the American Mathematical Society, vol. 107, no. 2, pp. 300–308, 1963.

7 S. Abdul Halim, “On a class of functions of complex order,”Tamkang Journal of Mathematics, vol. 30, no. 2, pp. 147–153, 1999.

8 W. C. Ma and D. Minda, “A unified treatment of some special classes of univalent functions,” in Proceedings of the Conference on Complex Analysis (Tianjin, 1992), Conf. Proc. Lecture Notes Anal., I, pp. 157–169, International Press, 1994.

9 V. Ravichandran, Y. Polatoglu, M. Bolcal, and A. Sen, “Certain subclasses of starlike and convex functions of complex order,”Hacettepe Journal of Mathematics and Statistics, vol. 34, pp. 9–15, 2005. 10 T. N. Shanmugam and S. Sivasubramanian, “On the Fekete-Szeg ¨o problem for some subclasses of

analytic functions,”Journal of Inequalities in Pure and Applied Mathematics, vol. 6, no. 3, article 71, pp. 1–6, 2005.

11 K. Suchithra, B. A. Stephen, and S. Sivasubramanian, “A coefficient inequality for certain classes of analytic functions of complex order,”Journal of Inequalities in Pure and Applied Mathematics, vol. 7, no. 4, article 145, pp. 1–6, 2006.

12 V. Karunakaran, “On a class of meromorphic starlike functions in the unit disc,” Mathematical Chronicle, vol. 4, no. 2-3, pp. 112–121, 1976.

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14 S. Ruscheweyh, “New criteria for univalent functions,”Proceedings of the American Mathematical Society, vol. 49, no. 1, pp. 109–115, 1975.

15 N. E. Cho, “Argument estimates of certain meromorphic functions,”Communications of the Korean Mathematical Society, vol. 15, no. 2, pp. 263–274, 2000.

16 K. S. Padmanabhan, “On certain subclasses of meromorphic functions in the unit disk,”Indian Journal of Pure and Applied Mathematics, vol. 30, no. 7, pp. 653–665, 1999.

17 M. R. Ganigi and B. A. Uralegaddi, “New criteria for meromorphic univalent functions,”Bulletin Math´ematique de la Soci´et´e des Sciences Math´ematiques de la R´epublique Socialiste de Roumanie, vol. 3381, no. 1, pp. 9–13, 1989.

18 B. A. Uralegaddi and M. D. Ganigi, “A new criterion for meromorphic convex functions,”Tamkang Journal of Mathematics, vol. 19, no. 1, pp. 43–48, 1988.

19 B. A. Uralegaddi and C. Somanatha, “Certain subclasses of meromorphic convex functions,”Indian Journal of Mathematics, vol. 32, no. 1, pp. 49–57, 1990.

20 W. G. Atshan and S. R. Kulkarni, “Subclass of meromorphic functions with positive coefficients defined by Ruscheweyh derivative—I,”Journal of Rajasthan Academy of Physical Sciences, vol. 6, no. 2, pp. 129–140, 2007.

21 N. E. Cho, “On certain subclasses of meromorphically multivalent convex functions,”Journal of Mathematical Analysis and Applications, vol. 193, no. 1, pp. 255–263, 1995.

References

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