Journal of Inequalities in Pure and
Applied Mathematics
COEFFICIENT INEQUALITY FOR A FUNCTION WHOSE DERIVATIVE
HAS A POSITIVE REAL PART
volume 7, issue 2, article 50,
2006.
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
Received 07 March, 2005;
accepted 09 March, 2006.
Communicated by: A. Sofo
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
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Abstract
Let R denote the
of normalised analytic univalent functions f defined
P∞subclass
n
by f(z) = z + n=2 an z and satisfy
Re{f 0 (z)} > 0
where z ∈ D = {z : |z| < 1}. The object of the present paper is to introduce the
functional |a2 a4 − a23 |. For f ∈ R, we give sharp upper bound for |a2 a4 − a23 |.
Coefficient Inequality For A
Function Whose Derivative Has
A Positive Real Part
2000 Mathematics Subject Classification: Primary 30C45.
Key words: Fekete-Szegö functional, Hankel determinant, Convex and starlike functions, Positive real functions.
Aini Janteng, Suzeini Abdul Halim
and Maslina Darus
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
Title Page
3
5
6
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J. Ineq. Pure and Appl. Math. 7(2) Art. 50, 2006
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1.
Introduction
Let A denote the class of normalised analytic functions f of the form
(1.1)
f (z) =
∞
X
an z n ,
n=0
where z ∈ D = {z : |z| < 1}. In [9], Noonan and Thomas stated that the qth
Hankel determinant of f is defined for q ≥ 1 by
an
an+1 · · · an+q+1 an+1 an+2 · · · an+q+2 Hq (n) = .
..
..
..
..
.
.
.
.
an+q−1 an+q · · · an+2q−2 Coefficient Inequality For A
Function Whose Derivative Has
A Positive Real Part
Aini Janteng, Suzeini Abdul Halim
and Maslina Darus
Title Page
Now, let S denote the subclass of A consisting of functions f of the form
(1.2)
f (z) = z +
∞
X
an z n
n=2
which are univalent in D.
A classical theorem of Fekete and Szegö [1] considered the Hankel determinant of f ∈ S for q = 2 and n = 1,
a1 a2 .
H2 (1) = a2 a3 Contents
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They made an early study for the estimates of |a3 − µa22 | when a1 = 1 and µ
real. The well-known result due to them states that if f ∈ S, then

if µ ≥ 1,

 4µ − 3,




−2µ
2
|a3 − µa2 | ≤
1 + 2 exp 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
class C of convex functions. Keogh and Merkes [5] obtained sharp estimates for
|a3 − µa22 | when f is close-to-convex, starlike and convex in D.
Here, we consider the Hankel determinant of f ∈ S for q = 2 and n = 2,
a2 a3 .
H2 (2) = a3 a4 Now, we are working on the functional
a sharp upper bound for the functional
is defined as the following.
|a2 a4 − a23 |.
|a2 a4 − a23 |
In this earlier work, we find
for f ∈ R. The subclass R
Definition 1.1. Let f be given by (1.2). Then f ∈ R if it satisfies the inequality
Coefficient Inequality For A
Function Whose Derivative Has
A Positive Real Part
Aini Janteng, Suzeini Abdul Halim
and Maslina Darus
Title Page
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(1.3)
0
Re{f (z)} > 0,
(z ∈ D).
The subclass R 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 + c1 z + c2 z 2 + · · ·
for z ∈ D.
Lemma 2.1 ([10]). If p ∈ P then |ck | ≤ 2 for each k.
Lemma 2.2 ([2]). The power series for p(z) given in (2.1) converges in D to a
function in P if and only if the Toeplitz determinants
2
c1
c2
· · · cn c−1
2
c1
· · · cn−1 (2.2)
Dn = ..
n = 1, 2, 3, . . .
..
..
..
.. ,
.
.
.
.
.
c−n c−n+1 c−n+2 · · ·
2 and
Pm c−k = c̄kit, are all nonnegative. They are strictly positive except for p(z) =
k
k=1 ρk p0 (e z), ρk > 0, tk real and tk 6= tj for k 6= j; in this case Dn > 0
for n < m − 1 and Dn = 0 for n ≥ m.
This necessary and sufficient condition is due to Carathéodory and Toeplitz
and can be found in [2].
Coefficient Inequality For A
Function Whose Derivative Has
A Positive Real Part
Aini Janteng, Suzeini Abdul Halim
and Maslina Darus
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3.
Main Result
Theorem 3.1. Let f ∈ R. Then
4
|a2 a4 − a23 | ≤ .
9
The result obtained is sharp.
Proof. We refer to the method by Libera and Zlotkiewicz [6, 7]. Since f ∈ R,
it follows from (1.3) that
(3.1)
f 0 (z) = p(z)
for some z ∈ D. Equating coefficients in (3.1) yields

2a2 = c1





3a3 = c2 .
(3.2)





4a4 = c3
From (3.2), it can be easily established that
c1 c3 c22 2
− .
|a2 a4 − a3 | = 8
9
c2 We make use of Lemma 2.2 to obtain the proper bound on c18c3 − 92 . We may
assume without restriction that c1 > 0. We begin by rewriting (2.2) for the cases
Coefficient Inequality For A
Function Whose Derivative Has
A Positive Real Part
Aini Janteng, Suzeini Abdul Halim
and Maslina Darus
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n = 2 and n = 3.
2 c1 c2 D2 = c1 2 c1 = 8 + 2 Re{c21 c2 } − 2|c2 |2 − 4c21 ≥ 0,
c̄2 c1 2 which is equivalent to
(3.3)
2c2 = c21 + x(4 − c21 )
Coefficient Inequality For A
Function Whose Derivative Has
A Positive Real Part
for some x, |x| ≤ 1. Then D3 ≥ 0 is equivalent to
|(4c3 − 4c1 c2 + c31 )(4 − c21 ) + c1 (2c2 − c21 )2 | ≤ 2 4 − c21
2
2
− 2 2c2 − c21 ;
Aini Janteng, Suzeini Abdul Halim
and Maslina Darus
and this, with (3.3), provides the relation
Title Page
(3.4)
4c3 = c31 + 2(4 − c21 )c1 x − c1 (4 − c21 )x2 + 2(4 − c21 )(1 − |x|2 )z,
for some value of z, |z| ≤ 1.
Suppose, now, that c1 = c and c ∈ [0, 2]. Using (3.3) along with (3.4) we get
c1 c3 c22 8 − 9
4
2
2
2
2 2
2
2
c
c
(4
−
c
)x
(4
−
c
)(32
+
c
)x
c(4
−
c
)(1
−
|x|
)z
+
−
+
= 288
144
288
16
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and an application of the triangle inequality shows that
c1 c3 c22 (3.5)
8 − 9
c4
c(4 − c2 ) c2 (4 − c2 )ρ (c − 2)(c − 16)(4 − c2 )ρ2
≤
+
+
+
288
16
144
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] and c ∈ [0, 2], then
F 0 (ρ) =
c2 (4 − c2 ) (c − 2)(c − 16)(4 − c2 )ρ
+
.
144
144
We note that F 0 (ρ) > 0 and consequently F is increasing and M axρ F (ρ) =
F (1), which contadicts our assumption of having the maximum value at the
interior point of ρ ∈ [0, 1]. Now let
c4
c(4 − c2 ) c2 (4 − c2 ) (c − 2)(c − 16)(4 − c2 )
G(c) = F (1) =
+
+
+
,
288
16
144
288
Coefficient Inequality For A
Function Whose Derivative Has
A Positive Real Part
Aini Janteng, Suzeini Abdul Halim
and Maslina Darus
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then
−c(5 + c2 )
G0 (c) =
=0
36
implies c = 0 which is a contradiction. Observe that
−5 − 3c2
G00 (c) =
< 0.
36
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Thus any maximum points of G must be on the boundary of c ∈ [0, 2]. However,
G(c) ≥ G(2) and thus G has maximum value at c = 0. The upper bound for
(3.5) corresponds to ρ = 1 and c = 0, in which case
c1 c3 c22 4
8 − 9 ≤ 9.
Equality is attained for functions in R given by
f 0 (z) =
1 + z2
.
1 − z2
This concludes the proof of our theorem.
Coefficient Inequality For A
Function Whose Derivative Has
A Positive Real Part
Aini Janteng, Suzeini Abdul Halim
and Maslina Darus
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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. GRENANDER AND G. 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. LIBERA AND E.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 mean p-valent functions, Trans. Amer. Math. Soc., 223(2)
(1976), 337–346.
Coefficient Inequality For A
Function Whose Derivative Has
A Positive Real Part
Aini Janteng, Suzeini Abdul Halim
and Maslina Darus
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[10] CH. POMMERENKE, Univalent Functions, Vandenhoeck and Ruprecht,
Göttingen, (1975)
Coefficient Inequality For A
Function Whose Derivative Has
A Positive Real Part
Aini Janteng, Suzeini Abdul Halim
and Maslina Darus
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