The purpose of the present paper is to establish some results concerning the partial sums of meromorphic starlike and meromorphic convex functions analogous to the results due to H

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Volume 5, Issue 2, Article 30, 2004

PARTIAL SUMS OF CERTAIN MEROMORPHIC FUNCTIONS

NAK EUN CHO AND SHIGEYOSHI OWA DEPARTMENT OFAPPLIEDMATHEMATICS

PUKYONGNATIONALUNIVERSITY

PUSAN608-737, KOREA.

necho@pknu.ac.kr DEPARTMENT OFMATHEMATICS

KINKIUNIVERSITY

HIGASHI-OSAKA, OSAKA577-8502 JAPAN.

owa@math.kindai.ac.jp

Received 03 March, 2004; accepted 30 March, 2004 Communicated by H.M. Srivastava

ABSTRACT. The purpose of the present paper is to establish some results concerning the partial sums of meromorphic starlike and meromorphic convex functions analogous to the results due to H. Silverman [J. Math. Anal. Appl. 209 (1997), 221-227]. Furthermore, we consider the partial sums of certain integral operator.

Key words and phrases: Partial sums, Meromorphic starlike functions, Meromorphic convex functions, Meromorphic close- to-convex functions, Integral operators.

2000 Mathematics Subject Classification. Primary 30C45.

1. INTRODUCTION

LetΣbe the class consisting of functions of the form

(1.1) f(z) = 1

z +

∞

X

k=1

a_kz^k

which are analytic in the punctured open unit disk D = {z : 0 < |z| < 1}. Let Σ^∗(α)and Σk(α)be the subclasses ofΣconsisting of all functions which are, respectively, meromorphic starlike and meromorphic convex of orderα (0≤ α < 1)in D. We also denote byΣ_c(α)the subclass ofΣwhich satisfies

−Re{z²f⁰(z)} > α (0≤α <1; z ∈ U =D ∪ {0}).

ISSN (electronic): 1443-5756

c 2004 Victoria University. All rights reserved.

This work was supported by the Korea Research Foundation Grant (KRF-2003-015-C00024).

048-04

We note that every function belonging to the class Σ_c(α) is meromorphic close-to-convex of order α inD (see [2]). If f(z) = P∞

k=0a_kz^k andg(z) = P∞

k=0b_kz^k are analytic in U, then their Hadamard product (or convolution), denoted byf∗g, is the function defined by the power series

(f ∗g)(z) =

∞

X

k=0

a_kb_kz^k (z ∈ U).

A sufficient condition for a functionf of the form (1.1) to be inΣ^∗(α)is that (1.2)

∞

X

k=1

(k+α)|ak| ≤1−α

and to be inΣ_k(α)is that (1.3)

∞

X

k=1

k(k+α)|ak| ≤1−α.

Further, we note that these sufficient conditions are also necessary for functions of the form (1.1) with positive or negative coefficients ([6, 13], also see [7]). Recently, Silverman [10]

determined sharp lower bounds on the real part of the quotients between the normalized starlike or convex functions and their sequences of partial sums. Also, Li and Owa [4] obtained the sharp radius which for the normalized univalent functions in U, the partial sums of the well- known Libera integral operator [5] imply starlikeness. Further, for various other interesting developments concerning partial sums of analytic univalent functions, the reader may be (for examples) refered to the works of Brickman et al. [1], Sheil-Small [9], Silvia [11], Singh and Singh [12] and Yang and Owa [14].

Since to a certain extent the work in the meromorphic univalent case has paralleled that of the analytic univalent case, one is tempted to search results analogous to those of Silverman [10] for meromorphic univalent functions inD. In the present paper, motivated essentially by the work of Silverman [10], we will investigate the ratio of a function of the form (1.1) to its sequence of partial sumsfn(z) = ¹_z +Pn

k=1akz^k when the coefficients are sufficiently small to satisfy either condition (1.2) or (1.3). More precisely, we will determine sharp lower bounds forRe{f(z)/f_n(z)}, Re{f_n(z)/f(z)}, Re{f⁰(z)/f_n⁰(z)},and Re{f_n⁰(z)/f⁰(z)}. Further, we give a property for the partial sums of certain integral operators in connection with meromor- phic close-to-convex functions. In the sequel, we will make use of the well-known result that Re{(1 +w(z))/(1−w(z))} > 0 (z ∈ U) if and only ifw(z) = P∞

k=1c_kz^k satisfies the in- equality|w(z)| < |z|. Unless otherwise stated, we will assume that f is of the form (1.1) and its sequence of partial sums is denoted byf_n(z) = ¹_z +Pn

k=1a_kz^k. 2. MAINRESULTS

Theorem 2.1. Iff of the form (1.1) satisfies condition (1.2), then

Re

f(z) f_n(z)

≥ n+ 2α

n+ 1 +α (z ∈ U).

The result is sharp for everyn, with extremal function

(2.1) f(z) = 1

z + 1−α

n+ 1 +αzⁿ⁺¹ (n ≥0).

Proof. We may write

n+ 1 +α 1−α

f(z)

f_n(z)− n+ 2α n+ 1 +α

= 1 +Pn

k=1a_kz^k+1+ ^n+1+α_1−α P∞

k=n+1a_kz^k+1 1 +Pn

k=1a_kz^k+1 := 1 +A(z)

1 +B(z).

Set(1 +A(z))/(1 +B(z)) = (1 +w(z))/(1−w(z)), so that w(z) = (A(z)−B(z))/(2 + A(z) +B(z)). Then

w(z) =

n+1+α 1−α

P∞

k=n+1a_kz^k+1 2 + 2Pn

k=1akz^k+1+^n+1+α_1−α P∞

k=n+1akz^k+1 and

|w(z)| ≤

n+1+α 1−α

P∞

k=n+1|a_k| 2−2Pn

k=1|a_k| −^n+1+α_1−α P∞

k=n+1|a_k|. Now|w(z)| ≤1if and only if

2

n+ 1 +α 1−α

^∞ X

k=n+1

|a_k| ≤2−2

n

X

k=1

|a_k|,

which is equivalent to (2.2)

n

X

k=1

|ak|+n+ 1 +α 1−α

∞

X

k=n+1

|ak| ≤1.

It suffices to show that the left hand side of (2.2) is bounded above byP∞

k=1((k+α)/(1−α))|ak|, which is equivalent to

n

X

k=1

k+ 2α−1 1−α

|a_k|+

∞

X

k=n+1

k−n−1 1−α

|a_k| ≥0.

To see that the functionf given by (2.1) gives the sharp result, we observe forz = re^πi/(n+2) that

f(z)

f_n(z) = 1 + 1−α

n+ 1 +αzⁿ⁺² −→1− 1−α

n+ 1 +α = n+ 2α

n+ 1 +α whenr→1⁻.

Therefore we complete the proof of Theorem 2.1.

Theorem 2.2. Iff of the form (1.1) satisfies condition (1.3), then

Re

f(z) f_n(z)

≥ (n+ 2)(n+α)

(n+ 1)(n+ 1 +α) (z ∈ U).

The result is sharp for everyn, with extremal function

(2.3) f(z) = 1

z + 1−α

(n+ 1)(n+ 1 +α)zⁿ⁺¹ (n≥0).

Proof. We write

(n+ 1)(n+ 1 +α) 1−α

f(z)

fn(z) − (n+ 2)(n+α) (n+ 1)(n+ 1 +α)

= 1 +Pn

k=1akz^k+1+ (n+1)(n+1+α) 1−α

P∞

k=n+1akz^k+1 1 +Pn

k=1a_kz^k+1 := 1 +w(z)

1−w(z), where

w(z) =

(n+1)(n+1+α) 1−α

P∞

k=n+1a_kz^k+1 2 + 2Pn

k=1a_kz^k+1+(n+1)(n+1+α) 1−α

P∞

k=n+1a_kz^k+1. Now

|w(z)| ≤

(n+1)(n+1+α) 1−α

P∞

k=n+1|a_k| 2−2Pn

k=1|a_k| −(n+1)(n+1+α) 1−α

P∞

k=n+1|a_k| ≤1, if

(2.4)

n

X

k=1

|ak|+(n+ 1)(n+ 1 +α) 1−α

∞

X

k=n+1

|ak| ≤1.

The left hand side of (2.4) is bounded above byP∞

k=1(k(k+α)/(1−α))|a_k|if 1

1−α ( _n

X

k=1

(k(k+α)−(1−α)|a_k|+

∞

X

k=n+1

(k(k+α)−(n+ 1)(n+ 1 +α))|a_k| )

≥0,

and the proof is completed.

We next determine bounds forRe{fn(z)/f(z)}.

Theorem 2.3. (a) Iff of the form (1.1) satisfies condition (1.2), then Re

f_n(z) f(z)

≥ n+ 1 +α

n+ 2 (z ∈ U).

(b) Iff of the form (1.1) satisfies condition (1.3), then Re

f_n(z) f(z)

≥ (n+ 1)(n+ 1 +α)

(n+ 1)(n+ 2)−n(1−α) (z ∈ U).

Equalities hold in (a) and (b) for the functions given by (2.1) and (2.3), respectively.

Proof. We prove (a). The proof of (b) is similar to (a) and will be omitted. We write n+ 2

1−α

f_n(z)

f(z) − n+ 1 +α n+ 2

= 1 +Pn

k=1akz^k+1+ ^n+1+α_1−α P∞

k=n+1akz^k+1 1 +Pn

k=1a_kz^k+1 := 1 +w(z)

1−w(z), where

|w(z)| ≤

n+2 1−α

P∞

k=n+1|a_k| 2−2Pn

k=1|a_k| − ^n+2α_1−α P∞

k=n+1|a_k| ≤1.

This last inequality is equivalent to (2.5)

n

X

k=1

|a_k|+n+ 1 +α 1−α

∞

X

k=n+1

|a_k| ≤1.

Since the left hand side of (2.5) is bounded above byP∞

k=1((k+α)/(1−α))|a_k|, the proof is

completed.

We turn to ratios involving derivatives. The proof of Theorem 2.4 below follows the pattern of those in Theorem 2.1 and (a) of Theorem 2.3 and so the details may be omitted.

Theorem 2.4. Iff of the form (1.1) satisfies condition (1.2) withα= 0, then

(a) Re

f⁰(z) f_n⁰(z)

≥ 0 (z ∈ U),

(b) Re

f_n⁰(z) f⁰(z)

≥ 1

2 (z ∈ U).

In both cases, the extremal function is given by (2.1) withα= 0.

Theorem 2.5. Iff of the form (1.1) satisfies condition (1.3), then

(a) Re

f⁰(z) f_n⁰(z)

≥ n+ 2α

n+ 1 +α (z ∈ U),

(b) Re

f_n⁰(z) f⁰(z)

≥ n+ 1 +α

n+ 2 (z ∈ U).

In both cases, the extremal function is given by (2.3).

Proof. It is well known thatf ∈ Σ_k(α) ⇔ −zf⁰ ∈ Σ^∗(α). In particular,f satisfies condition (1.3) if and only if −zf⁰ satisfies condition (1.2). Thus, (a) is an immediate consequence of Theorem 2.1 and (b) follows directly from (a) of Theorem 2.3.

For a functionf ∈Σ, we define the integral operatorF as follows:

F(z) = 1 z²

Z z

0

tf(t)dt= 1 z +

∞

X

k=1

1

k+ 2akz^k (z ∈ D).

Then-th partial sumF_nof the integral operatorF is given by F_n(z) = 1

z +

n

X

k=1

1

k+ 2a_kz^k (z ∈ D).

The following lemmas will be required for the proof of Theorem 2.8 below.

Lemma 2.6. For0≤θ≤π,

1 2+

m

X

k=1

cos(kθ) k+ 1 ≥ 0.

Lemma 2.7. LetP be analytic inU withP(0) = 1andRe{P(z)} > ¹₂ inU. For any function Qanalytic inU, the functionP ∗Qtakes values in the convex hull of the image onU underQ.

Lemma 2.6 is due to Rogosinski and Szegö [8] and Lemma 2.7 is a well-known result (c.f.

[3, 12]) that can be derived from the Herglotz’ representation forP. Finally, we derive

Theorem 2.8. Iff ∈Σ_c(α), thenF_n∈Σ_c(α)

Proof. Let f be of the form (1.1) and belong to the class Σ_c(α) for 0 ≤ α < 1. Since

−Re{z²f⁰(z)}> α, we have

(2.6) Re

(

1− 1

2(1−α)

∞

X

k=1

ka_kz^k+1 )

> 1

2 (z ∈ U).

Applying the convolution properties of power series toF_n⁰, we may write

−z²F_n⁰(z) = 1−

n

X

k=1

k

k+ 2a_kz^k+1 (2.7)

= 1− 1

2(1−α)

∞

X

k=1

ka_kz^k+1

!

∗ 1 + 2(1−α)

n+1

X

k=1

1 k+ 1z^k

! . Putting z = re^iθ(0 ≤ r < 1, 0 ≤ |θ| ≤ π), and making use of the minimum principle for harmonic functions along with Lemma 2.6, we obtain

Re (

1 + 2(1−α)

n+1

X

k=1

1 k+ 1z^k

)

= 1 + 2(1−α)

n+1

X

k=1

r^kcoskθ k+ 1 (2.8)

>1 + 2(1−α)

n+1

X

k=1

coskθ k+ 1 ≥α.

In view of (2.6), (2.7), (2.8) and Lemma 2.7, we deduce that

−Re{z²F_n⁰(z)} > α (0≤α <1; z ∈ U).

Therefore we complete the proof of Theorem 2.8.

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