# Hankel determinant problem of a subclass of analytic functions

## Abstract

In this article, we study the Hankel determinant problem of a subclass of analytic functions introduced recently by Arif et al.

2010 Mathematics Subject Classification: 30C45; 30C10.

## 1 Introduction

Let $\mathcal{A}$ be the class of analytic function satisfying the condition f (0) = 0, f'(0) - 1 = 0 in the open unit disc $\mathcal{E}=\left\{z:\mid z\mid <1\right\}$. By $\mathcal{S}$, ${\mathcal{S}}^{*}$, $\mathcal{C}$, and $\mathcal{K}$ we means the well-known subclasses of $\mathcal{A}$ which consist of univalent, starlike, convex, and close-to-convex functions, respectively.

Let ${\mathcal{V}}_{k}^{\lambda }\left(\sigma \right),\phantom{\rule{2.77695pt}{0ex}}k\ge 2,0\le \sigma <1,\phantom{\rule{2.77695pt}{0ex}}\lambda \phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{real}},\phantom{\rule{2.77695pt}{0ex}}\mid \lambda \mid <\frac{\pi }{2}$, denote the class of functions f1(z) analytic and locally univalent in , f1(0) = 0, ${f}_{1}^{\prime }\left(0\right)=1$ and satisfying

$\underset{0}{\overset{2\pi }{\int }}\mid \left\{\mathsf{\text{Re}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{e}^{i\lambda }\frac{{\left(z{f}_{1}^{\prime }\left(z\right)\right)}^{\prime }}{{f}_{1}^{\prime }\left(z\right)}-\sigma \mathsf{\text{cos}}\lambda \right\}/\left(1-\sigma \right)\mid \phantom{\rule{2.77695pt}{0ex}}d\theta \le k\pi \mathsf{\text{cos}}\lambda ,\phantom{\rule{2.77695pt}{0ex}}z=r{e}^{i\theta }.$
(1.1)

This class was introduced and studied in details by Moulis . For λ = 0, we obtain the class ${\mathcal{V}}_{k}\left(\sigma \right)$ of analytic functions with bounded boundary rotations of order σ studied by Padmanabhan et al.  and when σ = 0 and λ = 0, we get the class ${\mathcal{V}}_{k}$ discussed by Paatero , see also . Also it can easily be shown that ${f}_{1}\left(z\right)\in {\mathcal{V}}_{k}^{\lambda }\left(\sigma \right)$ if and only if there exists ${f}_{2}\left(z\right)\in {\mathcal{V}}_{k}$ such that

${f}_{1}^{\prime }\left(z\right)={{\left({f}_{2}^{\prime }\left(z\right)\right)}^{\left(1-\sigma \right)}}^{{e}^{-i\lambda }\mathsf{\text{cos}}\lambda }.$
(1.2)

We now consider a class of analytic functions defined by Arif et al.  as follows:

Definition 1.1. Let $f\left(z\right)\in \mathcal{A}$ in E. Then $f\left(z\right)\in {\stackrel{̃}{\mathcal{B}}}_{k}\left(\lambda ,\phantom{\rule{0.3em}{0ex}}\sigma ,\beta ,\gamma \right)$, if for k ≥ 2, 0 ≤ β ≤ 1, 0 ≤ γ ≤ 1, λ is real with $\mid \lambda \mid <\frac{\pi }{2}$ there exists a function ${f}_{1}\left(z\right)\in {\mathcal{V}}_{k}^{\lambda }\left(\sigma \right)$, 0 ≤ σ < 1, such that

$|\mathrm{arg}\left\{\frac{{z}^{1-\gamma }{f}^{\prime }\left(z\right)}{f\left(z\right)}{\left(\frac{f\left(z\right)}{{{f}^{\prime }}_{1}\left(z\right)}\right)}^{\gamma }\right\}|\le \frac{\beta \pi }{2},z\in E.$

By giving specific values to the parameters k, σ, λ, β, and γ in ${\stackrel{̃}{\mathcal{B}}}_{k}\left(\lambda ,\phantom{\rule{0.3em}{0ex}}\sigma ,\beta ,\gamma \right)$, we obtain many important subclasses studied by various authors in earlier articles, see .

Using (1.1) and (1.2), we have

$z{f}^{\prime }\left(z\right)={z}^{\gamma }{\left(f\left(z\right)\right)}^{1-\gamma }{\left({f}_{1}^{\prime }\left(z\right)\right)}^{\gamma }{p}^{\beta }\left(z\right),$
(1.3)

where ${f}_{1}\left(z\right)\in {\mathcal{V}}_{k}^{\lambda }\left(\sigma \right)$ and p(z) belongs to the class $\mathcal{P}$ of functions whose real part is positive.

Throughout in this article, we shall assume, unless otherwise stated, that k ≥ 2, 0 ≤ β ≤ 1, 0 < γ ≤ 1, λ is real with $\mid \lambda \mid <\frac{\pi }{2}$, 0 ≤ σ < 1.

In , the q th Hankel determinant ${\mathcal{H}}_{q}\left(n\right)$, q ≥ 1, n ≥ 1, for a function $f\left(z\right)\in \mathcal{A}$ is stated by Noonan and Thomas as:

Definition 1.2. Let $f\left(z\right)\in \mathcal{A}$. Then the q th Hankel determinant of f (z) is defined for q ≥ 1, n ≥ 1 by

${H}_{q}\left(n\right)=\mid \begin{array}{cccc}\hfill {a}_{n}\hfill & \hfill {a}_{n+1}\hfill & \hfill \cdots \phantom{\rule{0.3em}{0ex}}\hfill & \hfill {a}_{n+q-1}\hfill \\ \hfill {a}_{n+1}\hfill & \hfill {a}_{n+2}\hfill & \hfill \cdots \phantom{\rule{0.3em}{0ex}}\hfill & \hfill {a}_{n+q-2}\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill \\ \hfill {a}_{n+q-1}\hfill & \hfill {a}_{n+q-2}\hfill & \hfill \cdots \phantom{\rule{0.3em}{0ex}}\hfill & \hfill {a}_{n+2q-2}\hfill \end{array}\mid .$
(1.4)

The Hankel determinant plays an important role, for instance, in the study of the singularities by Hadamard, see [, p. 329], Edrei  and in the study of power series with integral coefficients by Polya [, p. 323], Cantor , and many others.

In this article, we shall determine the rate of growth of the Hankel determinant ${\mathcal{H}}_{q}\left(n\right)$ for $f\left(z\right)\in {\stackrel{̃}{\mathcal{B}}}_{k}\left(\lambda ,\sigma ,\beta ,\gamma \right)$with 0 < β < 2, as n → ∞. This determinant has been considered by several authors. That is, Noor  determined the rate of growth of ${\mathcal{H}}_{q}\left(n\right)$ as n → ∞ for a function f(z) belongs to the class ${\mathcal{V}}_{k}$. Pommerenke in  studied the Hankel determinant for starlike functions. The Hankel determinant problem for other interesting classes of analytic functions were discussed by Noor [11, 12, 24].

Lemma 1.1. Let $f\left(z\right)\in \mathcal{A}$. Let the q th Hankel determinant of f (z) for q ≥ 1, n ≥ 1 be defined by (1.4). Then, writting Δ j (n) = Δ j (n, z1, f(z)), we have

${\mathcal{H}}_{q}\left(n\right)=\mid \begin{array}{cccc}\hfill {\mathrm{\Delta }}_{2q-2}\left(n\right)\hfill & \hfill {\mathrm{\Delta }}_{2q-3}\left(n+1\right)\hfill & \hfill \cdots \phantom{\rule{0.3em}{0ex}}\hfill & \hfill {\mathrm{\Delta }}_{q-1}\left(n+q-1\right)\hfill \\ \hfill {\mathrm{\Delta }}_{2q-3}\left(n+1\right)\hfill & \hfill {\mathrm{\Delta }}_{2q-4}\left(n+2\right)\hfill & \hfill \cdots \phantom{\rule{0.3em}{0ex}}\hfill & \hfill {\mathrm{\Delta }}_{q-2}\left(n+q-2\right)\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill \\ \hfill {\mathrm{\Delta }}_{q-1}\left(n+q-1\right)\hfill & \hfill {\mathrm{\Delta }}_{q-2}\left(n+q-2\right)\hfill & \hfill \cdots \phantom{\rule{0.3em}{0ex}}\hfill & \hfill {\mathrm{\Delta }}_{q}\left(n+2q-2\right)\hfill \end{array}\mid ,$

where with Δ0(n) = a n , we define for j ≥ 1,

${\mathrm{\Delta }}_{j}\left(n,{z}_{1},f\left(z\right)\right)={\mathrm{\Delta }}_{j-1}\left(n,{z}_{1},f\left(z\right)\right)-{\mathrm{\Delta }}_{j-1}\left(n+1,{z}_{1},f\left(z\right)\right).$
(1.5)

Lemma 1.2. With ${z}_{1}=\frac{n}{n+1}y$ and v ≥ 0 any integer,

${\mathrm{\Delta }}_{j}\left(n+v,{z}_{1},z{f}^{\prime }\left(z\right)\right)=\sum _{m=0}^{j}\left(\begin{array}{c}j\\ m\end{array}\right)\frac{{y}^{m}\left(v-\left(m-1\right)n\right)}{{\left(n+1\right)}^{m}}{\mathrm{\Delta }}_{j-m}\left(n+m+v,f\left(z\right)\right).$

Lemmas 1.1 and 1.2 are due to Noonan and Thomas .

Lemma 1.3. Let h1(z) be starlike univalent function in . Then

1. (i)

there exists a z1 with |z1| = r such that for all z, |z| = r

$\mid z-{z}_{1}\mid \phantom{\rule{0.3em}{0ex}}\mid {h}_{1}\left(z\right)\mid \phantom{\rule{0.3em}{0ex}}\le \frac{2{r}^{2}}{1-{r}^{2}},$

see 

(ii)

$\frac{r}{{\left(1+r\right)}^{2}}\le \mid {h}_{1}\left(z\right)\mid \le \frac{r}{{\left(1+r\right)}^{2}},$

see.

## 2 Hankel determinant problem

Theorem 2.1. Let $f\left(z\right)\in {\stackrel{̃}{\mathcal{B}}}_{k}\left(\lambda ,\sigma ,\beta ,\gamma \right)$with 0 < β < 2 and let the q th Hankel determinant ${\mathcal{H}}_{q}\left(n\right)$ of f(z) be defined as in (1.4). Then

${\mathcal{H}}_{q}\left(n\right)=\mathsf{\text{O}}\phantom{\rule{0.3em}{0ex}}\left(1\right)\phantom{\rule{0.3em}{0ex}}{\left(M\left(r\right)\right)}^{1-\gamma }\left\{\begin{array}{c}{n}^{\gamma \left(\frac{k}{2}+1\right)\left(1-\sigma \right){\mathsf{\text{cos}}}^{2}\lambda +\beta -2},\phantom{\rule{0.3em}{0ex}}q=1\\ {n}^{\left\{\gamma \left(\frac{k}{2}+1\right)\left(1-\sigma \right){\mathsf{\text{cos}}}^{2}\lambda +\beta -1\right\}q-{q}^{2}},q\ge 2,\phantom{\rule{0.3em}{0ex}}k\ge \frac{8\left(q-1\right)}{\left(1-\sigma \right)\phantom{\rule{0.3em}{0ex}}\gamma {\mathsf{\text{cos}}}^{2}\lambda }-2,\end{array}\right\$

where $k>\frac{4j+2-\beta }{\left(1-\sigma \right)\gamma {\mathsf{\text{cos}}}^{2}\lambda }-2$ and O(1) is a constant depending on k, λ, β, σ, γ, and j only.

Proof. It is well known  that for starlike functions h1(z) and h2(z)

${f}_{1}^{\prime }\left(z\right)={\left[\frac{{\left({h}_{1}\left(z\right)/z\right)}^{\left(\frac{k}{4}+\frac{1}{2}\right)}}{{\left({h}_{2}\left(z\right)/z\right)}^{\left(\frac{k}{4}-\frac{1}{2}\right)}}\right]}^{\left(1-\sigma \right)\gamma {e}^{-i\lambda }\mathsf{\text{cos}}\lambda }.$
(2.1)

Using (2.1) in (1.3), we have

$z{f}^{\prime }\left(z\right)={z}^{\gamma }{\left(f\left(z\right)\right)}^{1-\gamma }{\left[\frac{{\left({h}_{1}\left(z\right)/z\right)}^{\left(\frac{k}{4}+\frac{1}{2}\right)}}{{\left({h}_{2}\left(z\right)/z\right)}^{\left(\frac{k}{4}-\frac{1}{2}\right)}}\right]}^{\left(1-\sigma \right)\gamma {e}^{-i\lambda }\mathsf{\text{cos}}\lambda }{p}^{\beta }\left(z\right).$
(2.2)

where $p\left(z\right)\in \mathcal{P}$.

Let F (z) = zf'(z). Then for j ≥ 1, z1 any non-zero complex and z = re, consider Δ j (n, z1, F(z)) as defined by (1.5). Then

$\mid {\mathrm{\Delta }}_{j}\left(n,{z}_{1},F\left(z\right)\right)\mid =\frac{1}{2\pi {r}^{n+j}}\mid \underset{0}{\overset{2\pi }{\int }}{\left(z-{z}_{1}\right)}^{j}F\left(z\right){e}^{i\left(n+j\right)\theta }d\theta \mid ,$

and by using (2.2), we have

$\begin{array}{c}\mid {\mathrm{\Delta }}_{j}\left(n,{z}_{1},F\left(z\right)\right)\mid \phantom{\rule{0.3em}{0ex}}\le \frac{1}{2\pi {r}^{n+j}}\underset{0}{\overset{2\pi }{\int }}\mid z-{z}_{1}{\mid }^{j}\mid z{\mid }^{\gamma }\mid f\left(z\right){\mid }^{1-\gamma }\left|{\left[\frac{{\left({h}_{1}\left(z\right)/z\right)}^{\left(\frac{k}{4}+\frac{1}{2}\right)}}{{\left({h}_{2}\left(z\right)/z\right)}^{\left(\frac{k}{4}-\frac{1}{2}\right)}}\right]}^{\left(1-\sigma \right)\gamma {e}^{-i\lambda }\mathsf{\text{cos}}\lambda }\right|\mid p\left(z\right){\mid }^{\beta }d\theta \\ \le \frac{{\left(M\left(r\right)\right)}^{1-\gamma }}{2\pi {r}^{n+j}}\underset{0}{\overset{2\pi }{\int }}\mid z-{z}_{1}{\mid }^{j}\left|{\left[\frac{{\left({h}_{1}\left(z\right)/z\right)}^{\left(\frac{k}{4}+\frac{1}{2}\right)}}{{\left({h}_{2}\left(z\right)/z\right)}^{\left(\frac{k}{4}-\frac{1}{2}\right)}}\right]}^{\left(1-\sigma \right)\gamma {e}^{-i\lambda }\mathsf{\text{cos}}\lambda }\right|\mid p\left(z\right){\mid }^{\beta }d\theta .\end{array}$
(2.3)

Since

$\begin{array}{cc}\hfill \left|{\left[\frac{{\left({h}_{1}\left(z\right)/z\right)}^{\left(\frac{k}{4}+\frac{1}{2}\right)}}{{\left({h}_{2}\left(z\right)/z\right)}^{\left(\frac{k}{4}-\frac{1}{2}\right)}}\right]}^{\left(1-\sigma \right)\gamma {e}^{-i\lambda }\mathsf{\text{cos}}\lambda }\right|& \le {\left[\frac{{\mid \left({h}_{1}\left(z\right)/z\right)\mid }^{\left(\frac{k}{4}+\frac{1}{2}\right)}}{{\mid \left({h}_{2}\left(z\right)/z\right)\mid }^{\left(\frac{k}{4}-\frac{1}{2}\right)}}\right]}^{\gamma \left(1-\sigma \right){\mathsf{\text{cos}}}^{2}\lambda }{e}^{\gamma \left(1-\sigma \right)\gamma \frac{\pi }{2}\frac{\mathsf{\text{sin}}2\lambda }{2}}\hfill \\ =\frac{{\mid \left({h}_{1}\left(z\right)/z\right)\mid }^{\left(\frac{k}{4}+\frac{1}{2}\right)\gamma \left(1-\sigma \right){\mathsf{\text{cos}}}^{2}\lambda }}{{\mid \left({h}_{2}\left(z\right)/z\right)\mid }^{\left(\frac{k}{4}-\frac{1}{2}\right)\gamma \left(1-\sigma \right){\mathsf{\text{cos}}}^{2}\lambda }}{c}_{1},\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{say}}\hfill \end{array}$

therefore (2.3) becomes

$\mid {\mathrm{\Delta }}_{j}\left(n,\phantom{\rule{2.77695pt}{0ex}}{z}_{1},\phantom{\rule{2.77695pt}{0ex}}F\left(z\right)\right)\mid \le \frac{{c}_{1}{\left(M\left(r\right)\right)}^{1-\gamma }}{2\pi {r}^{n+j}}\underset{0}{\overset{2\pi }{\int }}{\mid z-{z}_{1}\mid }^{j}\frac{{\mid \left({h}_{1}\left(z\right)/z\right)\mid }^{\left(\frac{k}{4}+\frac{1}{2}\right)\gamma \left(1-\sigma \right){\mathsf{\text{cos}}}^{2}\lambda }}{{\mid \left({h}_{2}\left(z\right)/z\right)\mid }^{\left(\frac{k}{4}-\frac{1}{2}\right)\gamma \left(1-\sigma \right){\mathsf{\text{cos}}}^{2}\lambda }}{\mid p\left(z\right)\mid }^{\beta }d\theta .$

Now using Lemma 1.3, we have

$\begin{array}{cc}\hfill \mid {\mathrm{\Delta }}_{j}\left(n,\phantom{\rule{2.77695pt}{0ex}}{z}_{1},\phantom{\rule{2.77695pt}{0ex}}F\left(z\right)\right)\mid & \le \frac{{c}_{1}{\left(M\left(r\right)\right)}^{1-\gamma }}{2\pi {r}^{n+j}}{\left(\frac{{\left(1+r\right)}^{2}}{r}\right)}^{\left(\frac{k}{4}-\frac{1}{2}\right)\gamma \left(1-\sigma \right){\mathsf{\text{cos}}}^{2}\lambda }\hfill \\ \frac{1}{{r}^{\gamma \left(1-\sigma \right){\mathsf{\text{cos}}}^{2}\lambda }}{\left(\frac{2{r}^{2}}{1-{r}^{2}}\right)}^{j}\underset{0}{\overset{2\pi }{\int }}\mid \left({h}_{1}\left(z\right)\right){\mid }^{\left(\frac{k}{4}+\frac{1}{2}\right)\gamma \left(1-\sigma \right){\mathsf{\text{cos}}}^{2}\lambda -j}\mid p\left(z\right){\mid }^{\beta }d\theta .\hfill \end{array}$

The well-known Holder's inequality will give us

$\begin{array}{cc}\hfill \mid {\mathrm{\Delta }}_{j}\left(n,\phantom{\rule{2.77695pt}{0ex}}{z}_{1},\phantom{\rule{2.77695pt}{0ex}}F\left(z\right)\right)\mid & \le \frac{{c}_{1}{\left(M\left(r\right)\right)}^{1-\gamma }{2}^{\left(\frac{k}{4}-\frac{1}{2}\right)\gamma \left(1-\sigma \right){\mathsf{\text{cos}}}^{2}\lambda +j}}{{r}^{n-j+\left(\frac{k}{4}+\frac{1}{2}\right)\gamma \left(1-\sigma \right){\mathsf{\text{cos}}}^{2}\lambda }}{\left(\frac{1}{1-r}\right)}^{j}{\left(\frac{1}{2\pi }\underset{0}{\overset{2\pi }{\int }}{\mid p\left(z\right)\mid }^{2}d\theta \right)}^{\frac{\beta }{2}}\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{\left(\frac{1}{2\pi }{\underset{0}{\overset{2\pi }{\int }}|\left({h}_{1}\left(z\right)\right)|}^{\frac{\left(\frac{k}{2}+1\right)\gamma \left(1-\sigma \right){\mathsf{\text{cos}}}^{2}\lambda -2j}{2-\beta }}d\theta \right)}^{\frac{2-\beta }{2}}.\hfill \end{array}$
(2.4)

Also, it is known  that, for $p\left(z\right)\in \mathcal{P}$, z E,

$\frac{1}{2\pi }\underset{0}{\overset{2\pi }{\int }}{\mid p\left(z\right)\mid }^{2}d\theta \le \frac{1+3{r}^{2}}{1-{r}^{2}}.$
(2.5)

Using (2.5) in (2.4), we obtain

$\begin{array}{cc}\hfill \mid {\mathrm{\Delta }}_{j}\left(n,\phantom{\rule{2.77695pt}{0ex}}{z}_{1},\phantom{\rule{2.77695pt}{0ex}}F\left(z\right)\right)\mid & \le \frac{{c}_{1}{\left(M\left(r\right)\right)}^{1-\gamma }{2}^{\left(\frac{k}{4}-\frac{1}{2}\right)\gamma \left(1-\sigma \right){\mathsf{\text{cos}}}^{2}\lambda +j}}{{r}^{n-j+\left(\frac{k}{4}+\frac{1}{2}\right)\gamma \left(1-\sigma \right){\mathsf{\text{cos}}}^{2}\lambda }}{\left(\frac{1}{1-r}\right)}^{j}\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{\left(\frac{1+3{r}^{2}}{1-{r}^{2}}\right)}^{\frac{\beta }{2}}{\left(\frac{1}{2\pi }\underset{0}{\overset{2\pi }{\int }}|\left({h}_{1}\left(z\right)\right)|\frac{\left(\frac{k}{4}+1\right)\gamma \left(1-\sigma \right){\mathsf{\text{cos}}}^{2}\lambda -2j}{2-\beta }d\theta \right)}^{\frac{2-\beta }{2}}.\hfill \end{array}$

Therefore, we can write

$\begin{array}{c}\mid {\mathrm{\Delta }}_{j}\left(n,\phantom{\rule{2.77695pt}{0ex}}{z}_{1},\phantom{\rule{2.77695pt}{0ex}}F\left(z\right)\right)\mid \phantom{\rule{0.3em}{0ex}}\le \frac{{c}_{1}{\left(M\left(r\right)\right)}^{1-\gamma }{2}^{\left(\frac{k}{4}-\frac{1}{2}\right)\gamma \left(1-\sigma \right){\mathsf{\text{cos}}}^{2}\lambda +\beta +j}}{{r}^{n-j+\left(\frac{k}{4}+\frac{1}{2}\right)\gamma \left(1-\sigma \right){\mathsf{\text{cos}}}^{2}\lambda }}{\left(\frac{1}{1-r}\right)}^{j+\frac{\beta }{2}}\\ {\left[{r}^{\left(\frac{k}{4}+\frac{1}{2}\right)\gamma \left(1-\sigma \right){\mathsf{\text{cos}}}^{2}\lambda -j}\frac{1}{2\pi }\underset{0}{\overset{2\pi }{\int }}\frac{d\theta }{{\mid 1-r{e}^{i\theta }\mid }^{\frac{\left(k+2\right)\gamma \left(1-\sigma \right){\mathsf{\text{cos}}}^{2}\lambda -4j}{2-\beta }}}\right]}^{\frac{2-\beta }{2}}.\end{array}$

Now using a subordination result for starlike functions, we have

$\begin{array}{cc}\hfill \mid {\mathrm{\Delta }}_{j}\left(n,\phantom{\rule{2.77695pt}{0ex}}{z}_{1},\phantom{\rule{2.77695pt}{0ex}}F\left(z\right)\right)\mid & \le \frac{{c}_{2}{\left(M\left(r\right)\right)}^{1-\gamma }}{{r}^{n}}{\left(\frac{1}{1-r}\right)}^{j+\frac{\beta }{2}}{\left[{\left(\frac{1}{1-r}\right)}^{\frac{\gamma \left(k+2\right)\left(1-\sigma \right){\mathsf{\text{cos}}}^{2}\lambda -4j}{2-\beta }-1}\right]}^{\frac{2-\beta }{2}},\hfill \\ =\frac{{c}_{2}{\left(M\left(r\right)\right)}^{1-\gamma }}{{r}^{n}}{\left(\frac{1}{1-r}\right)}^{\gamma \left(\frac{k}{2}+1\right)\left(1-\sigma \right){\mathsf{\text{cos}}}^{2}\lambda +\beta -1-j},\hfill \end{array}$

where c2 is a constant depending on k, λ, β, σ, γ, j only and γ(k + 2) (1 - σ) cos2 λ - 4j > 2 - β.

Applying Lemma 1.2 and putting ${z}_{1}=\left(\frac{n}{n+1}\right){e}^{i{\theta }_{n}},\phantom{\rule{2.77695pt}{0ex}}\left(n\phantom{\rule{2.77695pt}{0ex}}\to \infty \right),\phantom{\rule{2.77695pt}{0ex}}r=1-\frac{1}{n}$, we have for $k\ge \frac{4j-\beta +2}{\left(1-\sigma \right)\gamma {\mathsf{\text{cos}}}^{2}\lambda }-2$,

$\mid {\mathrm{\Delta }}_{j}\phantom{\rule{2.77695pt}{0ex}}\left(n,\phantom{\rule{2.77695pt}{0ex}}{e}^{i{\theta }_{n}},\phantom{\rule{2.77695pt}{0ex}}f\left(z\right)\right)\mid =\mathsf{\text{O}}\left(1\right)\phantom{\rule{0.3em}{0ex}}{\left(M\left(r\right)\right)}^{1-\gamma }{n}^{\gamma \left(\frac{k}{2}+1\right)\left(1-\sigma \right){\mathsf{\text{cos}}}^{2}\lambda +\beta -j-2},$

where O(1) is a constant depending on k, λ, β, σ, γ, and j only.

We now estimate the rate of growth of ${\mathcal{H}}_{q}\left(n\right)$.

For q = 1, ${\mathcal{H}}_{q}\left(n\right)={a}_{n}={\mathrm{\Delta }}_{0}\left(n\right)$ and

${H}_{1}\left(n\right)\phantom{\rule{2.77695pt}{0ex}}={a}_{n}=\mathsf{\text{O}}\left(1\right)\phantom{\rule{0.3em}{0ex}}{\left(M\left(r\right)\right)}^{1-\gamma }{n}^{\gamma \left(\frac{k}{2}+1\right)\left(1-\sigma \right){\mathsf{\text{cos}}}^{2}\lambda +\beta -2}.$

For q ≥ 2, we use similar argument due to Noonan and Thomas  together with Lemma 1.1 to have

${\mathcal{H}}_{q}\left(n\right)=\mathsf{\text{O}}\left(1\right)\phantom{\rule{0.3em}{0ex}}{\left(M\left(r\right)\right)}^{1-\gamma }{n}^{\left[\gamma \left(\frac{k}{2}+1\right)\left(1-\sigma \right){\mathsf{\text{cos}}}^{2}\lambda +\beta -1\right]q-{q}^{2}},\phantom{\rule{2.77695pt}{0ex}}k\ge \frac{8\left(q-1\right)}{\left(1-\sigma \right)\gamma {\mathsf{\text{cos}}}^{2}\lambda }-2$

and O(1) depends only on k, λ, β, σ, γ, and j.

For choosing different values of λ, β, σ, and γ in Theorem 2.1, we obtain the following results discussed by Noor [10, 11] and Noor et al. .

Corollary 2.1. For λ = 0, β = 1, σ = 0, $f\left(z\right)\in {\mathcal{B}}_{k}\left(\gamma \right)$, where the class ${\mathcal{B}}_{k}\left(\gamma \right)$ was introduced by Noor et al.  and

${\mathcal{H}}_{q}\left(n\right)=\mathsf{\text{O}}\left(1\right){\left(M\left(r\right)\right)}^{1-\gamma }\left\{\begin{array}{c}{n}^{\gamma \left(\frac{k}{2}+1\right)-1},\phantom{\rule{2.77695pt}{0ex}}q=1\\ {n}^{\gamma \left(\frac{k}{2}+1\right)q-{q}^{2}},\phantom{\rule{2.77695pt}{0ex}}q\ge 2,\phantom{\rule{2.77695pt}{0ex}}k\ge \frac{8\left(q-1\right)}{\gamma }-2\end{array}\right\$

where O(1) is a constant depending only on k and γ.

Corollary 2.2. For $\lambda =0,\phantom{\rule{0.3em}{0ex}}\gamma =1,\phantom{\rule{0.3em}{0ex}}f\left(z\right)\in {\stackrel{̃}{\mathcal{T}}}_{k}\left(\beta ,\phantom{\rule{0.3em}{0ex}}\sigma \right)$ and

${\mathcal{H}}_{q}\left(n\right)=\mathsf{\text{O}}\left(1\right)\left\{\begin{array}{c}{n}^{\left(1-\sigma \right)\left(\frac{k}{2}+1\right)+\beta -2},\phantom{\rule{2.77695pt}{0ex}}q=1\\ {n}^{\left[\left(1-\sigma \right)\left(\frac{k}{2}+1\right)+\beta -1\right]q-{q}^{2}},\phantom{\rule{2.77695pt}{0ex}}q\ge 2,\phantom{\rule{2.77695pt}{0ex}}k\ge \frac{8\left(q-1\right)}{\left(1-\sigma \right)}-2\end{array}\right\$

where O(1) is a constant depends on k, σ, and β only.

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### Competing interests

The authors declare that they have no competing interests.

### Authors' contributions

MA, KIN, and MR jointly discussed and presented the ideas of this article. MA made the text file and all the communications regarding the manuscript. All authors read and approved the final manuscript.

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Arif, M., Noor, K.I. & Raza, M. Hankel determinant problem of a subclass of analytic functions. J Inequal Appl 2012, 22 (2012). https://doi.org/10.1186/1029-242X-2012-22

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• DOI: https://doi.org/10.1186/1029-242X-2012-22

### Keywords

• Robertson function
• strongly Bazilevic functions
• bounded boundary rotations
• Hankel determinant 