# Bounds of Hankel Determinant for a Class of Univalent Functions

Authors: Sarika Verma; Sushma Gupta; Sukhjit Singh
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

---

## Abstract

The authors study the coefficient condition for the classℌα defined as the family of analytic functions f,f(0)=0 and f′(0)=1, which satisfy ℜ[(1-α)f′(z)+α(1+(zf″(z))/(f′(z)))]>α,z∈E, where α is a real number.


---

## Body

## 1. Introduction

Let𝒜 be the class of functions of the following form:
(1.1)f(z)=z+∑n=2∞‍anzn,
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
(1.2)R(f′(z)g′(z))>0,z∈E.
For fixed real numbers α, let ℌα denote the family of functions f in 𝒜 which satisfy
(1.3)R[(1-α)f′(z)+α(1+zf′′(z)f′(z))]>α,z∈E.In 2005, V. Singh et al. [1] established that, for 0<α<1, functions in ℌα satisfy ℜf′(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
(1.4)Hq(n)=|anan+1⋯an+q-1an+1an+2⋯an+q⋮⋮⋮⋮an+q-1an+q⋯an+2(q-1)|.The determinant has been investigated by several authors with the subject of inquiry ranging from rate of growth ofHq(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 Szego¨ [3] considered the Hankel determinant of f∈S for q=2 and n=1(1.5)H2(1)=|a1a2a2a3|.The well-known result due to them states that iff∈S, then
(1.6)|a3-μa22|≤{3-4μifμ≤0,1+2exp⁡(-2μ1-μ)if0≤μ<1,4μ-3ifμ≥1,
where a1=1 and μ is a real number. In the present paper, we obtain a sharp bound for H2(2)=|a2a4-a32| when f∈ℌα.


## 2. Preliminary Results

We denote byP the family of all functions p(z) given by
(2.1)p(z)=1+c1z+c2z2+⋯
analytic in E for which ℜ{p(z)}>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
(2.2)Dn=|2c1c2⋯cnc-12c1⋯cn-1⋮⋮c-nc-n+1c-n+2⋯2|n=1,2,3… and c-k = c¯k are all nonnegative. They are strictly positive except for
(2.3)p(z)=∑k=1m‍ρkp0(eitkz),ρk>0,tkreal,
and tk≠tj for k≠j; in this case, Dn>0 for n<m-1 and Dn=0 for n≥m.Lemma 2.2 (See [5, 6]).
Letp∈P. Then
(2.4)2c2=c12+x(4-c12),4c3=c13+2xc1(4-c12)-x2c1(4-c12)+2z(1-|x|2)(4-c12)
for some x,z such that |x|≤1 and |z|≤1.


## 3. Main Result

Theorem 3.1.
Letα, 0≤α≤1, be a real number. If f∈ℌα, then
(3.1)|a2a4-a32|≤{49(1-α)2(1+α)20≤α≤α0,K(α)α0≤α≤1,
where α0=0.4276891324… is the root of the equation 10α3-5α2+12α-5=0 and
(3.2)K(α)=(1-α)272(1+α)2(1+2α)[32(1+2α)-12[10α3-5α2+12α-5]2[4α5-6α4-18α3+29α2-20α-1]].Proof.
Sincef∈ℌα, it follows from (1.3) that there exists a function p∈P such that
(3.3)(1-α)f′(z)+α(1+zf′′(z)f′(z))=α+(1-α)p(z).
Equating coefficients in (3.3) yields
(3.4)2a2=(1-α)c1,3a3(1+α)=(1-α)c2+α(1-α)2c12,4a4(1+2α)=(1-α)c3+3α(1-α)2(1+α)c1c2+α(1-α)3(2α-1)(1+α)c13.
Thus, we can easily establish that(3.5)|a2a4-a32|=(1-α)272(1+α)2(1+2α)×|{9(1+α)2c1c3-8(1+2α)c22-α(1-α)(11-5α)c12c2+α(1-α)2(2α2+α-9)c14}|.
Using (2.4), in view of Lemma 2.2, we obtain that
(3.6)|a2a4-a32|=(1-α)272(1+α)2(1+2α)|{c144[(9α2+2α+1)+4α(2α2+α-9)(1-α)2+2α(1-α)(11-5α)]+12(4-c12)c12x[(9α2+2α+1)+α(1-α)(11-5α)]+92(1+α)2(4-c12)c1(1-|x|2)z-14x2(4-c12)×[32(1+2α)+c12(9α2+2α+1)]c144}|.
Sincep∈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
(3.7)|a2a4-a32|≤(1-α)272(1+α)2(1+2α){c44[(9α2+2α+1)+4α(2α2+α-9)(1-α)2+2α(1-α)(11-5α)]+12(4-c2)c2ρ[(9α2+2α+1)+α(1-α)(11-5α)]+92(1+α)2(4-c2)c+14(4-c2)(c-2)ρ2×[c(9α2+2α+1)-16(2α+1)]c44}≡F(ρ).
DifferentiatingF(ρ), we get the following:
(3.8)F′(ρ)=(1-α)272(1+α)2(1+2α){12(4-c2)c2[(9α2+2α+1)+α(1-α)(11-5α)]+12(4-c2)(c-2)ρ[c(9α2+2α+1)-16(2α+1)]}.
Using elementary calculus, one can show thatF′(ρ)>0 for ρ>0. It implies that F is an increasing function, and, thus, the upper bound for F(ρ) corresponds to ρ=1, in which case
(3.9)F(ρ)≤(1-α)272(1+α)2(1+2α)×{c44[4α(2α2+α-9)(1-α)2-2(9α2+2α+1)]+c2[3(9α2+2α+1)+2α(1-α)(11-5α)-8(2α+1)]+32(2α+1)c44}≡G(c).
Then,(3.10)G′(c)=(1-α)272(1+α)2(1+2α)c{2c2[2α(2α2+α-9)(1-α)2-(9α2+2α+1)]+2[3(9α2+2α+1)+2α(1-α)(11-5α)-8(2α+1)]}.
SettingG′(c)=0, since 0≤c≤2, we have
(3.11)c0=-(10α3-5α2+12α-5)4α5-6α4-18α3+29α2-20α-1
provided α≥α0, where α0=0.4276891324… is the root of the equation 10α3-5α2+12α-5=0.Case 1.
When0≤α≤α0, then the maximum value of G(c) corresponds to c=0. Therefore, we have
(3.12)max⁡0≤c≤2G(c)=G(0)=49(1-α)2(1+α)2.Case 2.
Whenα0≤α≤1, the maximum value of G(c) corresponds to c=c0. Therefore, we have
(3.13)max⁡0≤c≤2G(c)=G(c0)=K(α),
where K(α) is given by (3.2). This completes the proof of the Theorem.Settingα=0 in above theorem, we get the following result of Janteng et al. [7].Corollary 3.2.
If an analytic functionf is such that ℜ{f′(z)}>0, z∈E, then
(3.14)|a2a4-a32|≤49.
The result is sharp.



---

*Source: 147842-2012-07-15.xml*