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volume 5, issue 2, article 30, 2004.

Received 03 March, 2004;

accepted 30 March, 2004.

Communicated by:H.M. Srivastava

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Journal of Inequalities in Pure and Applied Mathematics

PARTIAL SUMS OF CERTAIN MEROMORPHIC FUNCTIONS

NAK EUN CHO AND SHIGEYOSHI OWA

Department of Applied Mathematics Pukyong National University Pusan 608-737, KOREA.

EMail:necho@pknu.ac.kr Department of Mathematics Kinki University

Higashi-Osaka, Osaka 577-8502 JAPAN.

EMail:owa@math.kindai.ac.jp

c

2000Victoria University ISSN (electronic): 1443-5756 048-04

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Partial Sums Of Certain Meromorphic Functions Nak Eun Cho and Shigeyoshi Owa

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J. Ineq. Pure and Appl. Math. 5(2) Art. 30, 2004

http://jipam.vu.edu.au

Abstract

The purpose of the present paper is to establish some results concerning the partial sums of meromorphic starlike and meromorphic convex functions anal- ogous 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.

2000 Mathematics Subject Classification:Primary 30C45.

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

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

1. Introduction

LetΣbe the class consisting of functions of the form

(1.1) f(z) = 1

z +

∞

X

k=1

akz^k

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

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

We note that every function belonging to the classΣ_c(α)is meromorphic close- to-convex of orderαinD(see [2]). Iff(z) = P∞

k=0a_kz^kandg(z) =P∞ k=0b_kz^k are analytic in U, then their Hadamard product (or convolution), denoted by f ∗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)|a_k| ≤1−α

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and to be inΣ_k(α)is that (1.3)

∞

X

k=1

k(k+α)|a_k| ≤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 se- quences of partial sums. Also, Li and Owa [4] obtained the sharp radius which for the normalized univalent functions inU, the partial sums of the well-known Libera integral operator [5] imply starlikeness. Further, for various other in- teresting 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 in D. 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 sumsf_n(z) = ¹_z+Pn

k=1a_kz^kwhen the coefficients are sufficiently small to satisfy either condition (1.2) or (1.3). More precisely, we will determine sharp lower bounds for Re{f(z)/fn(z)}, Re{fn(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 meromorphic close-to-convex functions. In the sequel, we will make use of the well-known result thatRe{(1 +w(z))/(1−

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w(z))} > 0 (z ∈ U)if and only ifw(z) = P∞

k=1c_kz^k satisfies the inequality

|w(z)|<|z|. Unless otherwise stated, we will assume thatf is of the form (1.1) and its sequence of partial sums is denoted byf_n(z) = ¹_z +Pn

k=1a_kz^k.

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2. Main Results

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

Re

f(z) fn(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)

fn(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 thatw(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=1a_kz^k+1+^n+1+α_1−α P∞

k=n+1a_kz^k+1 and

|w(z)| ≤

n+1+α 1−α

P∞

k=n+1|a_k| 2−2Pn

k=1|ak| − ^n+1+α_1−α P∞

k=n+1|ak|.

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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

|a_k|+n+ 1 +α 1−α

∞

X

k=n+1

|a_k| ≤1.

It suffices to show that the left hand side of (2.2) is bounded above byP∞ k=1((k+

α)/(1−α))|a_k|, 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 for z =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 Theorem2.1.

Re

f(z) f_n(z)

≥ (n+ 2)(n+α)

(n+ 1)(n+ 1 +α) (z ∈ U).

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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)

f_n(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+1akz^k+1 2 + 2Pn

k=1akz^k+1+ (n+1)(n+1+α) 1−α

P∞

k=n+1akz^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

|a_k|+(n+ 1)(n+ 1 +α) 1−α

∞

X

k=n+1

|a_k| ≤1.

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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), respec- tively.

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

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write

n+ 2 1−α

f_n(z)

f(z) − n+ 1 +α n+ 2

= 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 +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 2.1and (a) of Theorem2.3and 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),

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(b) Re

f_n⁰(z) f⁰(z)

≥ 1

2 (z ∈ U).

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

(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 that f ∈ 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 Theorem2.1and (b) follows directly from (a) of Theorem2.3.

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+ 2a_kz^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 Theorem2.8below.

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Lemma 2.6. For0≤θ ≤π,

1 2 +

m

X

k=1

cos(kθ) k+ 1 ≥ 0.

Lemma 2.7. LetP be analytic in U withP(0) = 1andRe{P(z)} > ¹₂ inU. For any functionQanalytic 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. Letf be of the form (1.1) and belong to the classΣ_c(α)for0≤ 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

kakz^k+1

!

∗ 1 + 2(1−α)

n+1

X

k=1

1 k+ 1z^k

! .

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Putting z = re^iθ(0 ≤ r < 1, 0 ≤ |θ| ≤ π), and making use of the minimum principle for harmonic functions along with Lemma2.6, 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 Lemma2.7, we deduce that

−Re{z²F_n⁰(z)} > α (0≤α <1; z ∈ U).

Therefore we complete the proof of Theorem2.8.

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References

[1] L. BRICKMAN, D.J. HALLENBECK, T.H. MacGREGOR AND D.

WILKEN, Convex hulls and extreme points of families of starlike and con- vex mappings, Trans. Amer. Math. Soc., 185 (1973), 413–428.

[2] M.D. GANIGI AND B.A. URALEGADDI, Subclasses of meromorphic close-to-convex functions, Bull. Math. Soc. Sci. Math. R. S. Roumanie (N.

S.), 33(81) (1989), 105–109.

[3] A.W. GOODMAN, Univalent functions, Vol. I, Mariner Publ. Co., Tampa, Fl., 1983.

[4] J.L. LI AND S. OWA, On partial sums of the Libera integral operator, J.

Math. Anal. Appl., 213 (1997), 444–454.

[5] R.J. LIBERA, Somes classes of regular univalent functions, Proc. Amer.

Math. Soc., 16 (1965), 755–758.

[6] M.L. MOGRA, T.R. REDDY ANDO.P. JUNEJA, Meromorphic univalent functions, Bull. Austral. Math. Soc., 32 (1985), 161–176.

[7] M.L. MOGRA, Hadamard product of certain meromorphic univalent func- tions, J. Math. Anal. Appl., 157 (1991), 10–16.

[8] W. ROGOSINSKIANDG. SZEGÖ, Uber die abschimlte von potenzreihen die in ernein kreise be schranket bleiben, Math. Z., 28 (1928), 73–94.

[9] T. SHEIL-SMALL, A note on partial sums of convex schlicht functions, Bull. London Math. Soc., 2 (1970), 165–168.

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[10] H. SILVERMAN, Partial sums of starlike and convex functions, J. Math.

Anal. Appl., 209 (1997), 221–227.

[11] E.M. SILVIA, On partial sums of convex functions of orderα, Houston J.

Math., 11 (1985), 397–404.

[12] R. SINGH AND S. SINGH, Convolution properties of a class of starlike functions, Proc. Amer. Math. Soc., 106 (1989), 145–152.

[13] B.A. URALEGADDI AND M.D. GANIGI, Meromorphic convex func- tions with negative coefficients, J. Math. Res. & Exposition, 7 (1987), 21–

26.

[14] D. YANG AND S. OWA, Subclasses of certain analytic functions, Hokkaido Math. J., 32 (2003), 127–136.