# Upper bound of the third Hankel determinant for a class of analytic functions related with lemniscate of Bernoulli

- Mohsan Raza
^{1}Email author and - Sarfraz Nawaz Malik
^{2}

**2013**:412

https://doi.org/10.1186/1029-242X-2013-412

© Raza and Malik; licensee Springer 2013

**Received: **15 February 2013

**Accepted: **8 August 2013

**Published: **28 August 2013

## Abstract

In this paper, the upper bound of the Hankel determinant ${H}_{3}(1)$ for a subclass of analytic functions associated with right half of the lemniscate of Bernoulli ${({x}^{2}+{y}^{2})}^{2}-2({x}^{2}-{y}^{2})=0$ is investigated.

**MSC:**30C45, 30C50.

### Keywords

starlike functions subordination lemniscate of Bernoulli Toeplitz determinants Hankel determinants## 1 Introduction and preliminaries

*A*be the class of functions

*f*of the form

which are analytic in the open unit disk $E=\{z:|z|<1\}$. A function *f* is said to be subordinate to a function *g*, written as $f\prec g$, if there exists a Schwartz function *w* with $w(0)=0$ and $|w(z)|<1$ such that $f(z)=g(w(z))$. In particular, if *g* is univalent in *E*, then $f(0)=g(0)$ and $f(E)\subset g(E)$.

*P*denote the class of analytic functions

*p*normalized by

This class of functions was introduced by Sokół and Stankiewicz [1] and further investigated by some authors. For details, see [2, 3].

*q*th Hankel determinant defined as

where $n\ge 1$ and $q\ge 1$. The Hankel determinant plays an important role in the study of singularities; for instance, see [[5], p.329] and Edrei [6]. This is also important in the study of power series with integral coefficients [[5], p.323] and Cantor [7]. For the use of the Hankel determinant in the study of meromorphic functions, see [8], and various properties of these determinants can be found in [[9], Chapter 4]. It is well known that the Fekete-Szegö functional $|{a}_{3}-{a}_{2}^{2}|={H}_{2}(1)$. This functional is further generalized as $|{a}_{3}-\mu {a}_{2}^{2}|$ for some *μ* (real as well as complex). Fekete and Szegö gave sharp estimates of $|{a}_{3}-\mu {a}_{2}^{2}|$ for *μ* real and $f\in S$, the class of univalent functions. It is a very great combination of the two coefficients which describes the area problems posted earlier by Gronwall in 1914-15. Moreover, we also know that the functional $|{a}_{2}{a}_{4}-{a}_{3}^{2}|$ is equivalent to ${H}_{2}(2)$. The *q* th Hankel determinant for some subclasses of analytic functions was recently studied by Arif *et al.* [10] and Arif *et al.* [11]. The functional $|{a}_{2}{a}_{4}-{a}_{3}^{2}|$ has been studied by many authors, see [12–14]. Babalola [15] studied the Hankel determinant ${H}_{3}(1)$ for some subclasses of analytic functions. In the present investigation, we determine the upper bounds of the Hankel determinant ${H}_{3}(1)$ for a subclass of analytic functions related with lemniscate of Bernoulli by using Toeplitz determinants.

We need the following lemmas which will be used in our main results.

**Lemma 1.1** [16]

*Let*$p\in P$

*and of the form*(1.2).

*Then*

*When*$v<0$

*or*$v>1$,

*the equality holds if and only if*$p(z)$

*is*$\frac{1+z}{1-z}$

*or one of its rotations*.

*If*$0<v<1$,

*then the equality holds if and only if*$p(z)=\frac{1+{z}^{2}}{1-{z}^{2}}$

*or one of its rotations*.

*If*$v=0$,

*the equality holds if and only if*$p(z)=(\frac{1}{2}+\frac{\eta}{2})\frac{1+z}{1-z}+(\frac{1}{2}-\frac{\eta}{2})\frac{1-z}{1+z}$ ($0\le \eta \le 1$)

*or one of its rotations*.

*If*$v=1$,

*the equality holds if and only if*

*p*

*is the reciprocal of one of the functions such that the equality holds in the case of*$v=0$.

*Although the above upper bound is sharp*,

*when*$0<v<1$,

*it can improved as follows*:

*and*

**Lemma 1.2** [16]

*If*$p(z)=1+{p}_{1}z+{p}_{2}{z}^{2}+\cdots $

*is a function with positive real part in*

*E*,

*then for*

*v*

*a complex number*

*This result is sharp for the functions*

**Lemma 1.3** [17]

*Let*$p\in P$

*and of the form*(1.2).

*Then*

*for some*

*x*, $|x|\le 1$,

*and*

*for some* *z*, $|z|\le 1$.

## 2 Main results

Although we have discussed the Hankel determinant problem in the paper, the first two problems are specifically related with the Fekete-Szegö functional, which is a special case of the Hankel determinant.

**Theorem 2.1**

*Let*$f\in {\mathit{SL}}^{\ast}$

*and of the form*(1.1).

*Then*

*Furthermore*,

*for*$-\frac{3}{4}<\mu \le \frac{1}{4}$,

*and for*$\frac{1}{4}<\mu \le \frac{5}{4}$,

*These results are sharp*.

*Proof*If $f\in {\mathit{SL}}^{\ast}$, then it follows from (1.3) that

Now, using Lemma 1.1, we have the required result. □

where $\mathrm{\Phi}(z)=\frac{z(z+\eta )}{1+\eta z}$ with $0\le \eta \le 1$.

**Theorem 2.2**

*Let*$f\in {\mathit{SL}}^{\ast}$

*and of the form*(1.1).

*Then for a complex number*

*μ*,

*Proof*Since

□

For $\mu =1$, we have ${H}_{2}(1)$.

**Corollary 2.3**

*Let*$f\in {\mathit{SL}}^{\ast}$

*and of the form*(1.1).

*Then*

**Theorem 2.4**

*Let*$f\in {\mathit{SL}}^{\ast}$

*and of the form*(1.1).

*Then*

*Proof*From (2.2), (2.3) and (2.4), we obtain

*ρ*, we obtain

*ρ*, we have

□

**Theorem 2.5**

*Let*$f\in {\mathit{SL}}^{\ast}$

*and of the form*(1.1).

*Then*

*Proof*Since

*ρ*, we get

*ρ*, which contradicts our assumption; therefore, $max{F}_{1}(p,\rho )={F}_{1}(p,0)={G}_{1}(p)$. This implies that

for $p=0$. Therefore $p=0$ is a point of maximum. Hence, we get the required result. □

**Lemma 2.6**

*If the function*$f(z)=\stackrel{\mathrm{\infty}}{\sum _{n=1}}{a}_{n}{z}^{n}$

*belongs to the class*${\mathit{SL}}^{\ast}$,

*then*

*These estimations are sharp*. *The first three bounds were obtained by Sokół* [3]*and the bound for* $|{a}_{5}|$ *can be obtained in a similar way*.

**Theorem 2.7**

*Let*$f\in {\mathit{SL}}^{\ast}$

*and of the form*(1.1).

*Then*

*Proof*Since

□

## Declarations

### Acknowledgements

The authors would like to thank the referees for helpful comments and suggestions which improved the presentation of the paper.

## Authors’ Affiliations

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