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volume 4, issue 4, article 79, 2003.

Received 15 July, 2002;

accepted 10 June, 2003.

Communicated by:N.E. Cho

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Journal of Inequalities in Pure and Applied Mathematics

PARTIAL SUMS OF FUNCTIONS OF BOUNDED TURNING

JAY M. JAHANGIRI AND K. FARAHMAND

Kent State University Burton, Ohio 44021-9500, USA.

EMail:[email protected] University of Ulster,

Jordanstown, BT37 0QB, United Kingdom.

EMail:[email protected]

c

2000Victoria University ISSN (electronic): 1443-5756 080-02

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Partial Sums of Functions of Bounded Turning

Jay M. Jahangiri and K. Farahmand

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J. Ineq. Pure and Appl. Math. 4(4) Art. 79, 2003

http://jipam.vu.edu.au

Abstract

We determine conditions under which the partial sums of the Libera integral operator of functions of bounded turning are also of bounded turning.

2000 Mathematics Subject Classification:Primary 30C45; Secondary 26D05.

Key words: Partial Sums, Bounded Turning, Libera Integral Operator.

1. Introduction

LetAdenote the family of functionsf which are analytic in the open unit disk U ={z :|z|<1}and are normalized by

(1.1) f(z) =z+

∞

X

k=2

a_kz^k, z ∈ U.

For0 ≤ α < 1,letB(α)denote the class of functionsf of the form (1.1) so that <(f⁰)> αinU. The functions inB(α)are called functions of bounded turning (c.f. [3, Vol. II]). By the Nashiro-Warschowski Theorem (see e.g. [3, Vol. I]) the functions inB(α)are univalent and also close-to-convex inU.

Forf of the form (1.1), the Libera integral operatorF is given by F(z) = 2

z Z z

0

f(ζ)dζ =z+

∞

X

k=2

2

k+ 1a_kz^k.

Then-th partial sumsF_n(z)of the Libera integral operatorF(z)are given by F_n(z) =z+

n

X

k=2

2

k+ 1a_kz^k.

In [5] it was shown that iff ∈ Ais starlike of orderα, α = 0.294...,then so is the Libera integral operatorF.We also know that (see e.g. [1]), there are functions which are univalent or spiral-like in U so that their Libera integral operators are not univalent or spiral-like in U. Li and Owa [4] proved that if f ∈ A is univalent in U, then Fn(z) is starlike in|z| < ³₈. The number ³₈ is

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sharp. In this paper we make use of a result of Gasper [2] to provide a simple proof for the following theorem.

Theorem 1.1 (Main Theorem). If ¹₄ ≤ α < 1 and f ∈ B(α), then F_n ∈ B ^4α−1₃

.

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2. Preliminary Lemmas

To prove our Main Theorem, we shall need the following three lemmas. The first lemma is due to Gasper ([2, Theorem 1]) and the third lemma is a well- known and celebrated result (c.f. [3, Vol. I]) which can be derived from Her- glotz’s representation for positive real part functions.

Lemma 2.1. Letθ be a real number andmandkbe natural numbers. Then

(2.1) 1

3+

m

X

k=1

cos(kθ) k+ 2 ≥0.

Lemma 2.2. Forz∈ U we have

<

m

X

k=1

z^k k+ 2

!

>−1 3.

Proof. For0≤r <1and for0≤ |θ| ≤πwritez =re^iθ =r(cos(θ)+isin(θ)).

By DeMoivre’s law and the minimum principle for harmonic functions, we have

(2.2) <

m

X

k=1

z^k k+ 2

!

=

m

X

k=1

r^kcos(kθ) k+ 2 >

m

X

k=1

cos(kθ) k+ 2 .

Now by Abel’s lemma (c.f. Titchmarsh [6]) and condition (2.1) of Lemma2.1 we conclude that the right hand side of (2.2) is greater than or equal to ⁻¹₃ .

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Lemma 2.3. Let P(z)be analytic in U, P(0) = 1,and <(P(z)) > ¹₂ in U. For functionsQanalytic inU the convolution functionP∗Qtakes values in the convex hull of the image onU underQ.

The operator “∗” stands for the Hadamard product or convolution of two power seriesf(z) = P∞

k=1a_kz^kandg(z) = P∞

k=1b_kz^kdenoted by(f∗g)(z) = P∞

k=1a_kb_kz^k.

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3. Proof of the Main Theorem

Letf be of the form (1.1) and belong toB(α)for ¹₄ ≤α <1.Since<(f⁰(z))>

αwe have

(3.1) < 1 + 1

2(1−α)

∞

X

k=2

ka_kz^k−1

!

> 1 2.

Applying the convolution properties of power series toF_n⁰(z)we may write F_n⁰(z)

(3.2)

= 1 +

n

X

k=2

2k

k+ 1a_kz^k−1

= 1 + 1

2(1−α)

∞

X

k=2

ka_kz^k−1

!

∗ 1 + (1−α)

n

X

k=2

4 k+ 1z^k−1

!

=P(z)∗Q(z).

From Lemma2.2form =n−1we obtain

(3.3) <

n

X

k=2

z^k−1 k+ 1

!

>−1 3.

Applying a simple algebra to the above inequality (3.3) andQ(z)in (3.2) yields

<(Q(z)) =< 1 + (1−α)

n

X

k=2

4 k+ 1z^k−1

!

> 4α−1 3 .

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On the other hand, the power seriesP(z)in (3.2) in conjunction with the condition (3.1) yields<(P(z))> ¹₂.Therefore, by Lemma2.3, <(F_n⁰(z))> ^4α−1₃ . This concludes the Main Theorem.

Remark 3.1. The Main Theorem also holds forα < ¹₄.We also note thatB(α) forα <0is no longer a bounded turning family.

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References

[1] D.M. CAMPBELLANDV. SINGH, Valence properties of the solution of a differential equation, Pacific J. Math., 84 (1979), 29–33.

[2] G. GASPER, Nonnegative sums of cosines, ultraspherical and Jacobi poly- nomials, J. Math. Anal. Appl., 26 (1969), 60–68.

[3] A.W. GOODMAN, Univalent Functions, Vols. I & II, Mariner Pub. 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] P.T. MOCANU, M.O. READE AND D. RIPEANU, The order of starlike- ness of a Libera integral operator, Mathematica (Cluj), 19 (1977), 67–73.

[6] E.C. TITCHMARSH, The Theory of Functions, 2nd Ed., Oxford University Press, 1976.