Bounds of Hankel Determinant for a Class of Univalent Functions

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Volume 2012, Article ID 147842,6pages 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,[email protected] Received 31 March 2012; Revised 1 June 2012; Accepted 1 June 2012

Academic Editor: Teodor Bulboaca

Copyrightq2012 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 functionsf, f0 0 andf0 1, which satisfy1−αfzα1zfz/fz> α, z∈E, whereαis a real number.

1. Introduction

LetAbe the class of functions of the following form:

fz z^∞

n2

anzⁿ, 1.1

which are analytic in the unit discE{z:|z|<1}, and letSbe the subclass ofAconsisting of functions which are univalent inE. A functionf ∈ Ais said to be close to convex in the open unit discEif there exists a convex functiongnot necessarily normalizedsuch that

fz

gz

>0, z∈E. 1.2

For fixed real numbersα, letHαdenote the family of functionsfinAwhich satisfy

1−αfz α

1zfz fz

> α, z∈E. 1.3

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In 2005, V. Singh et al. 1established that, for 0 < α < 1, functions inHα satisfy fz>0 inEand so are close to convex inE.

In2, Noonan and Thomas defined the Hankel determinantHqnof the functionf forq≥1 andn≥1 by

Hqn

a_n a_n1 · · · a_nq−1 a_n1 a_n2 · · · a_nq

... ... ... ... a_nq−1 a_nq · · · a_n2q−1

. 1.4

The determinant has been investigated by several authors with the subject of inquiry ranging from rate of growth ofHqnasn → ∞, to the determination of precise bounds on H_qnfor specificqandnfor some special classes of functions. In a classical theorem, Fekete and Szeg ¨o3considered the Hankel determinant off∈Sforq2 andn1

H₂1 a₁ a₂

a2 a3

. 1.5

The well-known result due to them states that iff∈S, then

a₃−μa₂²≤

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

3−4μ if μ≤0,

12 exp −2μ

1−μ

if 0≤μ <1,

4μ−3 ifμ≥1,

1.6

where a₁ 1 and μ is a real number. In the present paper, we obtain a sharp bound for H22 |a2a4−a²₃|whenf∈Hα.

2. Preliminary Results

We denote byPthe family of all functionspzgiven by

pz 1c₁zc₂z²· · · 2.1

analytic inEfor which{pz}>0 forz∈E. It is well known that forp∈P,|ck| ≤2 for each k.

Lemma 2.1See4. The power series for p(z) given in2.1converges inEto a function inP if and only if the Toeplitz determinants

D_n

2 c1 c2 · · · cn

c₋₁ 2 c₁ · · · c_n−1

... ...

c_−n c_−n1 c_−n2 · · · 2

2.2

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n1,2,3. . .andc_−k=ckare all nonnegative. They are strictly positive except for

pz ^m

k1

ρkp0

e^it^kz

, ρk>0, tk real, 2.3

andtk/tjfork /j; in this case,Dn>0 forn < m−1 andDn0 forn≥m.

Lemma 2.2See5,6. Letp∈P. Then

2c₂c₁²x

4−c₁² ,

4c₃c₁³2xc₁

4−c₁²

−x²c₁ 4−c₁²

2z

1− |x|²

4−c₁² 2.4

for somex, zsuch that|x| ≤1 and|z| ≤1.

3. Main Result

Theorem 3.1. Letα, 0≤α≤1, be a real number. Iff∈Hα, then

a2a₄−a₃²≤

⎧⎪

⎨

⎪⎩ 4 9

1−α²

1α² 0≤α≤α0, Kα α₀≤α≤1,

3.1

whereα₀0.4276891324. . .is the root of the equation 10α³−5α²12α−50 and

Kα 1−α²

721α²12α

3212α−1 2

10α³−5α²12α−5₂ 4α⁵−6α⁴−18α³29α²−20α−1

. 3.2

Proof. Sincef ∈Hα, it follows from1.3that there exists a functionp∈Psuch that

1−αfz α

1 zfz fz

α 1−αpz. 3.3

Equating coeﬃcients in3.3yields

2a2 1−αc1,

3a31α 1−αc2α1−α²c12, 4a412α 1−αc33α1−α²

1α c1c2α1−α³2α−1 1α c13.

3.4

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Thus, we can easily establish that

a2a₄−a₃² 1−α²

721α²12α×

91α²c₁c₃−812αc22−α1−α11−5αc12c₂

α1−α²

2α²α−9 c₁⁴.

3.5

Using2.4, in view ofLemma 2.2, we obtain that a2a4−a32

1−α² 721α²12α

c₁⁴ 4

9α²2α1 4α

2α²α−9

1−α²2α1−α11−5α

1 2

4−c12 c12x

9α²2α1

α1−α11−5α 9

21α²

4−c₁² c₁

1− |x|² z−1

4x²

4−c₁²

×

3212α c12

9α²2α1 .

3.6

Sincep∈P, so|c1| ≤2. Lettingc₁c, we may assume without restriction thatc∈0,2.

Thus, applying the triangle inequality on3.6, withρ|x| ≤1, we obtain a₂a₄−a₃²

≤ 1−α² 721α²12α

c⁴ 4

9α²2α1 4α

2α²α−9

1−α²2α1−α11−5α

1 2

4−c² c²ρ

9α²2α1

α1−α11−5α 9

21α² 4−c²

c1 4

4−c²

c−2ρ²

× c

9α²2α1

−162α1

≡F ρ

.

3.7

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DiﬀerentiatingFρ, we get the following:

F ρ

1−α² 721α²12α

1 2

4−c² c²

9α²2α1

α1−α11−5α

1 2

4−c²

c−2ρ c

9α²2α1

−162α1 .

3.8

Using elementary calculus, one can show thatFρ>0 forρ >0. It implies thatFis an increasing function, and, thus, the upper bound forFρcorresponds toρ 1, in which case

F ρ

≤ 1−α² 721α²12α

× c⁴

4 4α

2α²α−9

1−α²−2

9α²2α1

c² 3

9α²2α1

2α1−α11−5α−82α1

322α1

≡Gc.

3.9

Then,

Gc 1−α²

721α²12αc 2c²

2α

2α²α−9

1−α²−

9α²2α1

2 3

9α²2α1

2α1−α11−5α−82α1 .

3.10

SettingGc 0, since 0≤c≤2, we have

c₀

−

10α³−5α²12α−5

4α⁵−6α⁴−18α³29α²−20α−1 3.11

providedα≥α₀, whereα₀0.4276891324. . .is the root of the equation 10α³−5α²12α−5 0.

Case 1. When 0≤α≤ α0, then the maximum value ofGccorresponds toc 0. Therefore, we have

max0≤c≤2Gc G0 4 9

1−α²

1α². 3.12

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Case 2. Whenα0 ≤ α≤ 1, the maximum value ofGccorresponds toc c0. Therefore, we have

max0≤c≤2Gc Gc0 Kα, 3.13

whereKαis given by3.2. This completes the proof of the Theorem.

Settingα0 in above theorem, we get the following result of Janteng et al.7.

Corollary 3.2. If an analytic functionfis such that{fz}>0,z∈E, then a2a4−a32≤ 4

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. Szeg ¨o, “Eine Bemerkung uber ungerade schlichte Funktionen,” Journal of the London Mathematical Society, vol. 8, no. 2, pp. 85–89.

4 U. Grenander and G. Szeg ¨o, 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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