COEFFICIENT BOUNDS FOR MEROMORPHIC STARLIKE AND CONVEX FUNCTIONS

(1)

Coefficient Bounds S. K. Lee, V. Ravichandran

and S. Shamani vol. 10, iss. 3, art. 71, 2009

Title Page

Contents

JJ II

J I

Page1of 14 Go Back Full Screen

Close

COEFFICIENT BOUNDS FOR MEROMORPHIC STARLIKE AND CONVEX FUNCTIONS

SEE KEONG LEE V. RAVICHANDRAN

Universiti Sains Malaysia Department of Mathematics

11800 USM Penang, University of Delhi

Malaysia Delhi 110 007, India

EMail:sklee@cs.usm.my EMail:vravi@maths.du.ac.in

SUPRAMANIAM SHAMANI

School of Mathematical Sciences Universiti Sains Malaysia 11800 USM Penang, Malaysia EMail:sham105@hotmail.com

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, Fekete- Szegö inequality.

(2)

Title Page Contents

JJ II

J I

Close 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. Let φbe a function with positive real part on∆ withφ(0) = 1,φ⁰(0) > 0 which maps∆onto a region starlike with respect to1which is symmetric with respect to the real axis. The classP∗

(p, w0, φ)consists of functions f ∈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)and P(p, φ)are obtained.

Acknowledgment: This research is supported by Short Term grant from Universiti Sains Malaysia and also a grant from University of Delhi.

(3)

Title Page Contents

JJ II

J I

Close

1. Introduction

LetU(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 inU^∗(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 inU^∗(p, w₀)all omit the valuew₀. This class of starlike meromorphic functions is developed from Robertson’s concept of star center points [11]. Let P denote the class of functionsP(z)which are meromorphic in∆and satisfyP(0) = 1 and<{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 allz ∈ ∆. LetP∗

(p, w₀)denote the class of functionsf(z)which satisfy (1.1) and the condition f(0) = 0, f⁰(0) = 1. ThenU^∗(p, w₀) is a subset ofP∗

(p, w₀).

Miller [9] proved thatU^∗(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 somep < 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.

(5)

Title Page Contents

JJ II

J I

Page5of 14 Go Back Full Screen

Close

Iff ∈K(p),then for eachzin∆,

(1.2) <

1 +zf⁰⁰(z)

f⁰(z) + 2p

z−p − 2pz 1−pz

≤0.

Let P

(p) denote the class of functions f which satisfy (1.2) and the conditions f(0) = 0andf⁰(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 quantitieszf⁰(z)/f(z)or1 +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 analytic functionφwith positive real part on ∆with φ(0) = 1, φ⁰(0) > 0which maps the unit disk∆onto a region starlike (univalent) with respect to1which is symmetric with respect to the real axis, they considered the classS^∗(φ)consisting of functionsf ∈ Afor whichzf⁰(z)/f(z)≺φ(z) (z ∈∆).

They also investigated a corresponding class C(φ) of functions f ∈ A satisfying 1 + 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. Let φ be a function with positive real part on ∆ with φ(0) = 1, φ⁰(0) >0which maps∆onto a region starlike with respect to1which is symmetric

(6)

Title Page Contents

JJ II

J I

Close

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

≺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.

(7)

JJ II

J I

Close

2. Coefficients Bound Problem

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. Let φ(z) = 1 +B₁z +B₂z² +· · · andf(z) = z +a₂z² +· · · in

|z|< p. Iff ∈P∗

(p, w₀, φ), then

w₀ = 2p

pB1c1−2p²−2 and

(2.1) p

p²+B1p+ 1 ≤ |w₀| ≤ p p²−B1p+ 1.

(8)

Title Page Contents

JJ II

J I

Close

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

w²₀ +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

(9)

Title Page Contents

JJ II

J I

Close

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.

(10)

JJ II

J I

Close

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 Lemma2.1.

The classes P∗

(p, w₀, φ) andP

(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 Theorem2.2.

Corollary 2.4 ([10, Theorem 1, p. 447]). Letf ∈P∗

(p, w0)andf(z) = z+a2z²+

· · · in|z|< p. Then the second coefficienta2 is 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 Theorem2.2.

The next theorem is for convex meromorphic functions.

(11)

Title Page Contents

JJ II

J I

Close

Theorem 2.5. Let φ(z) = 1 +B₁z +B₂z² +· · · andf(z) = z +a₂z² +· · · in

|z|< p. Iff ∈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 |^2B_B²

1 + 3µB₁| ≥2,

B1

6 if |^2B_B²

1 + 3µB₁| ≤2.

Proof. Lethnow be defined by h(z) =−

1 + zf⁰⁰(z)

f⁰(z) + 2p

= 1 +b1z+b2z²+· · ·

andp1 be defined as in the proof of Theorem2.2. 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 .

(12)

Title Page Contents

JJ II

J I

Close

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 Lemma2.1.

(13)

Title Page Contents

JJ II

J I

Close

References

[1] R.M. ALI, V. RAVICHANDRAN AND N. SEENIVASAGAN, Coefficient bounds forp-valent functions, Appl. Math. Comput., 187(1) (2007), 35–46.

[2] R.M. ALI, V. RAVICHANDRAN, AND S.K. LEE, Subclasses of multiva- lent 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 univa- lent 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. MAANDD. MINDA, A unified treatment of some special classes of univa- lent 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.

(14)

Title Page Contents

JJ II

J I

Close

[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. LEEANDV. RAVICHANDRAN, Convolution and differential subordination for multivalent functions, Bull. Malays. Math.

Sci. Soc. (2), 32(3) (2009), to appear.