Forf ∈ R, we give sharp upper bound for|a2a4−a23|

http://jipam.vu.edu.au/

Volume 7, Issue 2, Article 50, 2006

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

AINI JANTENG, SUZEINI ABDUL HALIM, AND MASLINA DARUS INSTITUTE OFMATHEMATICALSCIENCES

UNIVERSITIMALAYA, 50603 KUALALUMPUR, MALAYSIA

aini_jg@ums.edu.my

INSTITUTE OFMATHEMATICALSCIENCES

UNIVERSITIMALAYA

50603 KUALALUMPUR, MALAYSIA

suzeini@um.edu.my SCHOOL OFMATHEMATICALSCIENCES

FACULTY OFSCIENCES ANDTECHNOLOGY

UNIVERSITIKEBANGSAANMALAYSIA

43600 BANGI, SELANGOR, MALAYSIA

maslina@pkrisc.cc.ukm.my

Received 07 March, 2005; accepted 09 March, 2006 Communicated by A. Sofo

ABSTRACT. LetRdenote the subclass of normalised analytic univalent functionsf defined by f(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|.

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

2000 Mathematics Subject Classification. Primary 30C45.

1. INTRODUCTION

LetAdenote the class of normalised analytic functionsf of the form

(1.1) f(z) =

∞

X

n=0

a_nzⁿ,

ISSN (electronic): 1443-5756

c 2006 Victoria University. All rights reserved.

068-05

where z ∈ D = {z : |z| < 1}. In [9], Noonan and Thomas stated that the qth 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, letSdenote 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 ∈ Sfor q= 2andn= 1,

H₂(1) =

a₁ a₂ a₂ a₃

.

They made an early study for the estimates of|a₃ −µa²₂| whena₁ = 1and µreal. 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 ∈ S forq= 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 subclassRis 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 subclassRwas studied systematically by MacGregor [8] who indeed referred to numer- ous earlier investigations involving functions whose derivative has a positive real part.

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

2. PRELIMINARYRESULTS

LetP be the family of all functionspanalytic inDfor whichRe{p(z)}>0and (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 · · · cn

c₋₁ 2 c₁ · · · c_n−1 ... ... ... ... ... c_−n c_−n+1 c_−n+2 · · · 2

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

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

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

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

3. MAINRESULT

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 Lemma 2.2 to obtain the proper bound on

c1c3

8 − ^c₉²²

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

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 of ρ∈[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) > 0 and consequently F is increasing and M 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.

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

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. KEOGHANDE.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 func- tion, 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 inP, 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. NOONANANDD.K. THOMAS, On the second Hankel determinant of areally meanp-valent functions, Trans. Amer. Math. Soc., 223(2) (1976), 337–346.

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