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volume 7, issue 2, article 50, 2006.

Received 07 March, 2005;

accepted 09 March, 2006.

Communicated by:A. Sofo

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Journal of Inequalities in Pure and Applied Mathematics

COEFFICIENT INEQUALITY FOR A FUNCTION WHOSE DERIVATIVE HAS A POSITIVE REAL PART

AINI JANTENG, SUZEINI ABDUL HALIM AND MASLINA DARUS

Institute of Mathematical Sciences Universiti Malaya

50603 Kuala Lumpur, Malaysia.

EMail:aini_jg@ums.edu.my Institute of Mathematical Sciences Universiti Malaya

50603 Kuala Lumpur, Malaysia.

EMail:suzeini@um.edu.my School of Mathematical Sciences Faculty of Sciences and Technology Universiti Kebangsaan Malaysia 43600 Bangi, Selangor, Malaysia.

EMail:maslina@pkrisc.cc.ukm.my

c

2000Victoria University ISSN (electronic): 1443-5756 068-05

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Coefficient Inequality For A Function Whose Derivative Has

A Positive Real Part

Aini Janteng, Suzeini Abdul Halim and Maslina Darus

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J. Ineq. Pure and Appl. Math. 7(2) Art. 50, 2006

http://jipam.vu.edu.au

Abstract

LetRdenote the subclass of normalised analytic univalent functionsf defined byf(z) =z+P∞

n=2a_nzⁿand satisfy

Re{f⁰(z)}>0

wherez∈ D={z:|z|<1}. The object of the present paper is to introduce the functional|a₂a4−a²₃|. Forf∈ R, we give sharp upper bound for|a₂a4−a²₃|.

2000 Mathematics Subject Classification:Primary 30C45.

Key words: Fekete-Szegö functional, Hankel determinant, Convex and starlike func- tions, Positive real functions.

1. Introduction

LetAdenote the class of normalised analytic functionsf of the form

(1.1) f(z) =

∞

X

n=0

anzⁿ,

wherez ∈ D = {z : |z| <1}. In [9], Noonan and Thomas stated that theqth Hankel determinant off is defined forq ≥1by

H_q(n) =

a_n a_n+1 · · · a_n+q+1 a_n+1 a_n+2 · · · a_n+q+2

... ... ... ... an+q−1 a_n+q · · · an+2q−2

.

Now, letS denote the subclass ofAconsisting of functionsf of the form

(1.2) f(z) = z+

∞

X

n=2

a_nzⁿ

which are univalent inD.

A classical theorem of Fekete and Szegö [1] considered the Hankel determinant off ∈ Sforq= 2andn = 1,

H₂(1) =

a₁ a₂ a₂ a₃

.

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They made an early study for the estimates of |a₃ −µa²₂|when a₁ = 1and µ real. The well-known result due to them states that iff ∈ S, then

|a₃−µa²₂| ≤











4µ−3, if µ≥1,

1 + 2 exp_−2µ

1−µ

, if 0≤µ≤1, 3−4µ, if µ≤0.

Hummel [3, 4] proved the conjecture of V. Singh that |a₃ −a²₂| ≤ ¹₃ for the classC of convex functions. Keogh and Merkes [5] obtained sharp estimates for

|a₃−µa²₂|whenf is close-to-convex, starlike and convex inD.

Here, we consider the Hankel determinant off ∈ Sforq= 2andn= 2, H₂(2) =

a₂ a₃ a₃ a₄

.

Now, we are working on the functional|a₂a₄−a²₃|. In this earlier work, we find a sharp upper bound for the functional|a₂a₄−a²₃|forf ∈ R. The subclassR is defined as the following.

Definition 1.1. Letf be given by (1.2). Thenf ∈ Rif it satisfies the inequality

(1.3) Re{f⁰(z)}>0, (z ∈ D).

The subclassR was studied systematically by MacGregor [8] who indeed referred to numerous earlier investigations involving functions whose derivative has a positive real part.

We first state some preliminary lemmas which shall be used in our proof.

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2. Preliminary Results

Let P be the family of all functions p analytic in D for which Re{p(z)} > 0 and

(2.1) p(z) = 1 +c₁z+c₂z²+· · · forz ∈ D.

Lemma 2.1 ([10]). Ifp∈ P then|c_k| ≤2for eachk.

Lemma 2.2 ([2]). The power series forp(z)given in (2.1) converges inDto a function inP if and only if the Toeplitz determinants

(2.2) D_n =

2 c₁ c₂ · · · c_n c−1 2 c₁ · · · cn−1

... ... ... ... ... c−n c−n+1 c−n+2 · · · 2

, n= 1,2,3, . . .

andc−k = ¯ck, are all nonnegative. They are strictly positive except forp(z) = Pm

k=1ρ_kp₀(e^it^kz), ρ_k >0, t_k real andt_k 6= t_j fork 6= j; in this caseD_n > 0 forn < m−1andD_n = 0forn ≥m.

This necessary and sufficient condition is due to Carathéodory and Toeplitz and can be found in [2].

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3. Main Result

Theorem 3.1. Letf ∈R. Then

|a₂a₄ −a²₃| ≤ 4 9.

The result obtained is sharp.

Proof. We refer to the method by Libera and Zlotkiewicz [6,7]. Sincef ∈ R, it follows from (1.3) that

(3.1) f⁰(z) = p(z)

for somez ∈ D. Equating coefficients in (3.1) yields

(3.2)











2a2 =c1

3a₃ =c₂ 4a₄ =c₃

.

From (3.2), it can be easily established that

|a₂a₄−a²₃|=

c₁c₃ 8 −c²₂

9 .

We make use of Lemma2.2to obtain the proper bound on

c1c3

8 − ^c₉²²

. We may assume without restriction thatc1 >0. We begin by rewriting (2.2) for the cases

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n = 2andn = 3.

D₂ =

2 c₁ c₂ c₁ 2 c₁

¯

c₂ c₁ 2

= 8 + 2 Re{c²₁c₂} −2|c₂|² −4c²₁ ≥0,

which is equivalent to

(3.3) 2c₂ =c²₁+x(4−c²₁) for somex,|x| ≤1. ThenD₃ ≥0is equivalent to

|(4c₃−4c₁c₂+c³₁)(4−c²₁) +c₁(2c₂ −c²₁)²| ≤2 4−c²₁2

−2

2c₂−c²₁

2;

and this, with (3.3), provides the relation

(3.4) 4c3 =c³₁+ 2(4−c²₁)c1x−c1(4−c²₁)x²+ 2(4−c²₁)(1− |x|²)z, for some value ofz, |z| ≤1.

Suppose, now, thatc₁ =candc∈[0,2]. Using (3.3) along with (3.4) we get

c₁c₃ 8 − c²₂

9

=

c⁴

288 + c²(4−c²)x

144 − (4−c²)(32 +c²)x²

288 + c(4−c²)(1− |x|²)z 16

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and an application of the triangle inequality shows that

c₁c₃ 8 −c²₂

9 (3.5)

≤ c⁴

288 +c(4−c²)

16 + c²(4−c²)ρ

144 + (c−2)(c−16)(4−c²)ρ² 288

=F(ρ)

with ρ = |x| ≤ 1. We assume that the upper bound for (3.5) attains at the interior point ofρ∈[0,1]andc∈[0,2], then

F⁰(ρ) = c²(4−c²)

144 + (c−2)(c−16)(4−c²)ρ

144 .

We note that F⁰(ρ) > 0and consequentlyF is increasing andM ax_ρ F(ρ) = F(1), which contadicts our assumption of having the maximum value at the interior point ofρ∈[0,1]. Now let

G(c) =F(1) = c⁴

288 +c(4−c²)

16 +c²(4−c²)

144 +(c−2)(c−16)(4−c²)

288 ,

then

G⁰(c) = −c(5 +c²) 36 = 0 impliesc= 0which is a contradiction. Observe that

G⁰⁰(c) = −5−3c² 36 <0.

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Thus any maximum points ofGmust be on the boundary ofc∈[0,2]. However, G(c) ≥ G(2) and thusG has maximum value atc = 0. The upper bound for (3.5) corresponds toρ= 1andc= 0, in which case

c₁c₃ 8 −c²₂

9

≤ 4 9. Equality is attained for functions inRgiven by

f⁰(z) = 1 +z² 1−z². This concludes the proof of our theorem.

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References

[1] M. FEKETE AND G. SZEGÖ, Eine Bemerkung uber ungerade schlichte funktionen, J. London Math. Soc., 8 (1933), 85–89.

[2] U. GRENANDERANDG. SZEGÖ, Toeplitz Forms and their Application, Univ. of California Press, Berkeley and Los Angeles, (1958)

[3] J. HUMMEL, The coefficient regions of starlike functions, Pacific J.

Math., 7 (1957), 1381–1389.

[4] J. HUMMEL, Extremal problems in the class of starlike functions, Proc.

Amer. Math. Soc., 11 (1960), 741–749.

[5] F.R. KEOGH AND E.P. MERKES, A coefficient inequality for certain classes of analytic functions, Proc. Amer. Math. Soc., 20 (1969), 8–12.

[6] R.J. LIBERAANDE.J. ZLOTKIEWICZ, Early coefficients of the inverse of a regular convex function, Proc. Amer. Math. Soc., 85(2) (1982), 225–

230.

[7] R.J. LIBERA AND E.J. ZLOTKIEWICZ, Coefficient bounds for the inverse of a function with derivative in P, Proc. Amer. Math. Soc., 87(2) (1983), 251–289.

[8] T.H. MACGREGOR, Functions whose derivative has a positive real part, Trans. Amer. Math. Soc., 104 (1962), 532–537.

[9] J.W. NOONAN AND D.K. THOMAS, On the second Hankel determinant of areally meanp-valent functions, Trans. Amer. Math. Soc., 223(2) (1976), 337–346.

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[10] CH. POMMERENKE, Univalent Functions, Vandenhoeck and Ruprecht, Göttingen, (1975)