Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 147842, 6 pages
doi:10.1155/2012/147842
Research Article
Bounds of Hankel Determinant for
a Class of Univalent Functions
Sarika Verma, Sushma Gupta, and Sukhjit Singh
Department of Mathematics, Sant Longowal Institute of Engineering and Technology, Punjab,
Longowal 148106, India
Correspondence should be addressed to Sarika Verma, sarika.16984@gmail.com
Received 31 March 2012; Revised 1 June 2012; Accepted 1 June 2012
Academic Editor: Teodor Bulboaca
Copyright q 2012 Sarika Verma et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
The authors study the coefficient condition for the class Hα defined as the family of analytic
functions f, f0 0 and f 0 1, which satisfy 1−αf zα1zf z/f z > α, z ∈ E,
where α is a real number.
1. Introduction
Let A be the class of functions of the following form:
fz z ∞
an zn ,
1.1
n2
which are analytic in the unit disc E {z : |z| < 1}, and let S be the subclass of A consisting
of functions which are univalent in E. A function f ∈ A is said to be close to convex in the
open unit disc E if there exists a convex function g not necessarily normalized such that
f z
g z
> 0,
1.2
z ∈ E.
For fixed real numbers α, let Hα denote the family of functions f in A which satisfy
zf z
1 − αf z α 1 > α,
f z
z ∈ E.
1.3
2
International Journal of Mathematics and Mathematical Sciences
In 2005, V. Singh et al. 1 established that, for 0 < α < 1, functions in Hα satisfy
f z > 0 in E and so are close to convex in E.
In 2, Noonan and Thomas defined the Hankel determinant Hq n of the function f
for q ≥ 1 and n ≥ 1 by
an an1 · · · anq−1 a
n1 an2 · · · anq .
Hq n .
..
..
..
..
.
.
.
anq−1 anq · · · an2q−1 1.4
The determinant has been investigated by several authors with the subject of inquiry
ranging from rate of growth of Hq n as n → ∞, to the determination of precise bounds on
Hq n for specific q and n for some special classes of functions. In a classical theorem, Fekete
and Szegö 3 considered the Hankel determinant of f ∈ S for q 2 and n 1
a1 a2 .
H2 1 a2 a3 1.5
The well-known result due to them states that if f ∈ S, then
⎧
⎪
⎪3 − 4μ ⎨
⎪
−2μ
2
a3 − μa2 ≤ 1 2 exp
⎪
1−μ
⎪
⎪
⎩
4μ − 3
if μ ≤ 0,
if 0 ≤ μ < 1,
1.6
if μ ≥ 1,
where a1 1 and μ is a real number. In the present paper, we obtain a sharp bound for
H2 2 |a2 a4 − a23 | when f ∈ Hα .
2. Preliminary Results
We denote by P the family of all functions pz given by
pz 1 c1 z c2 z2 · · ·
2.1
analytic in E for which {pz} > 0 for z ∈ E. It is well known that for p ∈ P , |ck | ≤ 2 for each
k.
Lemma 2.1 See 4. The power series for p(z) given in 2.1 converges in E to a function in P if
and only if the Toeplitz determinants
2
c1
c2 · · · cn c
c1 · · · cn−1 −1 2
Dn .
.. ..
. c−n c−n1 c−n2 · · · 2 2.2
International Journal of Mathematics and Mathematical Sciences
3
n 1, 2, 3 . . . and c−k = ck are all nonnegative. They are strictly positive except for
pz m
ρk p0 eitk z ,
ρk > 0, tk real,
k1
2.3
and tk / j; in this case, Dn > 0 for n < m − 1 and Dn 0 for n ≥ m.
/ tj for k Lemma 2.2 See 5, 6. Let p ∈ P . Then
2c2 c1 2 x 4 − c1 2 ,
4c3 c1 2xc1 4 − c1
3
2
− x c1 4 − c1
2
2
2z 1 − |x|2 4 − c1 2
2.4
for some x, z such that |x| ≤ 1 and |z| ≤ 1.
3. Main Result
Theorem 3.1. Let α, 0 ≤ α ≤ 1, be a real number. If f ∈ Hα , then
⎧
2
⎨ 4 1 − α
⎪
a2 a4 − a3 2 ≤ 9 1 α2
⎪
⎩
Kα
0 ≤ α ≤ α0 ,
3.1
α0 ≤ α ≤ 1,
where α0 0.4276891324 . . . is the root of the equation 10α3 − 5α2 12α − 5 0 and
2
10α3 − 5α2 12α − 5
1
Kα 321 2α − 5
.
2 4α − 6α4 − 18α3 29α2 − 20α − 1
721 α2 1 2α
1 − α2
3.2
Proof. Since f ∈ Hα , it follows from 1.3 that there exists a function p ∈ P such that
zf z
α 1 − αpz.
1 − αf z α 1 f z
3.3
Equating coefficients in 3.3 yields
2a2 1 − αc1 ,
3a3 1 α 1 − αc2 α1 − α2 c1 2 ,
4a4 1 2α 1 − αc3 2
3.4
3
3α1 − α
α1 − α 2α − 1 3
c1 c2 c1 .
1 α
1 α
4
International Journal of Mathematics and Mathematical Sciences
Thus, we can easily establish that
a2 a4 − a3 2 1 − α2
721 α2 1 2α
× 91 α2 c1 c3 − 81 2αc2 2 − α1 − α11 − 5αc1 2 c2
α1 − α2 2α2 α − 9 c1 4 .
3.5
Using 2.4, in view of Lemma 2.2, we obtain that
a2 a4 − a3 2 c 4 1
2
2
2
9α
2α
1
4α
2α
α
−
9
2α1
−
α11
−
5α
−
α
1
721 α2 1 2α 4
1 − α2
1
4 − c1 2 c1 2 x 9α2 2α 1 α1 − α11 − 5α
2
9
1 1 α2 4 − c1 2 c1 1 − |x|2 z − x2 4 − c1 2
2
4
× 321 2α c1 2 9α2 2α 1
.
3.6
Since p ∈ P , so |c1 | ≤ 2. Letting c1 c, we may assume without restriction that c ∈ 0, 2.
Thus, applying the triangle inequality on 3.6, with ρ |x| ≤ 1, we obtain
a2 a4 − a3 2 c4 2
2
2
9α
≤
2α1
4α
2α
α
−
9
2α1
−
α11
−
5α
−
α
1
721 α2 1 2α 4
1 − α2
1
4 − c2 c2 ρ 9α2 2α 1 α1 − α11 − 5α
2
9
1
1α2 4 − c2 c 4 − c2 c − 2ρ2
2
4
2
× c 9α 2α 1 − 162α 1 ≡ F ρ .
3.7
International Journal of Mathematics and Mathematical Sciences
5
Differentiating Fρ, we get the following:
F ρ 1
4 − c2 c2 9α2 2α 1 α1 − α11 − 5α
2
721 α 1 2α 2
1 − α2
1
2
2
4 − c c − 2ρ c 9α 2α 1 − 162α 1 .
2
3.8
Using elementary calculus, one can show that F ρ > 0 for ρ > 0. It implies that F is
an increasing function, and, thus, the upper bound for Fρ corresponds to ρ 1, in which
case
F ρ ≤
1 − α2
721 α2 1 2α
c4 2
×
4α 2α α − 9 1 − α2 − 2 9α2 2α 1
4
c 3 9α2 2α 1 2α1 − α11 − 5α − 82α 1 322α 1
2
≡ Gc.
3.9
Then,
G c 1 − α2
721 α2 1 2α
c 2c2 2α 2α2 α − 9 1 − α2 − 9α2 2α 1
2 3 9α2 2α 1 2α1 − α11 − 5α − 82α 1 .
3.10
Setting G c 0, since 0 ≤ c ≤ 2, we have
c0 − 10α3 − 5α2 12α − 5
4α5 − 6α4 − 18α3 29α2 − 20α − 1
3.11
provided α ≥ α0 , where α0 0.4276891324 . . . is the root of the equation 10α3 − 5α2 12α − 5 0.
Case 1. When 0 ≤ α ≤ α0 , then the maximum value of Gc corresponds to c 0. Therefore,
we have
max Gc G0 0≤c≤2
4 1 − α2
.
9 1 α2
3.12
6
International Journal of Mathematics and Mathematical Sciences
Case 2. When α0 ≤ α ≤ 1, the maximum value of Gc corresponds to c c0 . Therefore, we
have
max Gc Gc0 Kα,
0≤c≤2
3.13
where Kα is given by 3.2. This completes the proof of the Theorem.
Setting α 0 in above theorem, we get the following result of Janteng et al. 7.
Corollary 3.2. If an analytic function f is such that {f z} > 0, z ∈ E, then
4
a2 a4 − a3 2 ≤ .
9
3.14
The result is sharp.
References
1 V. Singh, S. Singh, and S. Gupta, “A problem in the theory of univalent functions,” Integral Transforms
and Special Functions, vol. 16, no. 2, pp. 179–186, 2005.
2 J. W. Noonan and D. K. Thomas, “On the second Hankel determinant of areally mean p-valent
functions,” Transactions of the American Mathematical Society, vol. 223, pp. 337–346, 1976.
3 M. Fekete and G. Szegö, “Eine Bemerkung uber ungerade schlichte Funktionen,” Journal of the London
Mathematical Society, vol. 8, no. 2, pp. 85–89.
4 U. Grenander and G. Szegö, Toeplitz Forms and Their Applications, California Monographs in
Mathematical Sciences, University of California Press, Berkeley, Calif, USA, 1958.
5 R. J. Libera and E. J. Złotkiewicz, “Early coefficients of the inverse of a regular convex function,”
Proceedings of the American Mathematical Society, vol. 85, no. 2, pp. 225–230, 1982.
6 R. J. Libera and E. J. Złotkiewicz, “Coefficient bounds for the inverse of a function with derivative in
P,” Proceedings of the American Mathematical Society, vol. 87, no. 2, pp. 251–257, 1983.
7 A. Janteng, S. A. Halim, and M. Darus, “Coefficient inequality for a function whose derivative has a
positive real part,” Journal of Inequalities in Pure and Applied Mathematics, vol. 7, no. 2, article 50, pp. 1–5,
2006.
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