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 ${\mathcal{V}}_k^{\lambda }\left(\sigma \right),k\ge 2,0\le \sigma <1,\lambda \mathsf{\text{real}},\mid \lambda \mid <\frac{\pi }{2}$, denote the class of functions f₁(z) analytic and locally univalent in ℰ, f₁(0) = 0, $f_1^{\prime }\left(0\right)=1$ and satisfying

$$\underset{0}{\overset{2\pi }{\int }}\mid \left\{\mathsf{\text{Re}}{e}^{i\lambda }\frac{{\left(z{f}_1^{\prime }\left(z\right)\right)}^{\prime }}{f_1^{\prime }\left(z\right)}-\sigma \mathsf{\text{cos}}\lambda \right\}/\left(1-\sigma \right)\mid d\theta \le k\pi \mathsf{\text{cos}}\lambda ,z=r{e}^{i\theta }.$$

(1.1)

This class was introduced and studied in details by Moulis [1]. 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. [2] and when σ = 0 and λ = 0, we get the class ${\mathcal{V}}_k$ discussed by Paatero [3], see also [4–8]. 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. [9] as follows:

Definition 1.1. Let $f\left(z\right)\in \mathcal{A}$ in E. Then $f\left(z\right)\in {\tilde{\mathcal{B}}}_k\left(\lambda ,\sigma ,\beta ,\gamma \right)$, 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

$$\left|\arg \left\{\frac{z^{1-\gamma }{f}^{\prime }(z)}{f(z)}{\left(\frac{f(z)}{{f^{\prime }}_1(z)}\right)}^{\gamma }\right\}\right|\le \frac{\beta \pi }{2},z\in E.$$

By giving specific values to the parameters k, σ, λ, β, and γ in ${\tilde{\mathcal{B}}}_k\left(\lambda ,\sigma ,\beta ,\gamma \right)$, we obtain many important subclasses studied by various authors in earlier articles, see [10–16].

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 [17], 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

$$H_q\left(n\right)=\mid \begin{array}{cccc}\hfill a_n\hfill & \hfill a_{n+1}\hfill & \hfill \cdots \hfill & \hfill a_{n+q-1}\hfill \\ \hfill a_{n+1}\hfill & \hfill a_{n+2}\hfill & \hfill \cdots \hfill & \hfill a_{n+q-2}\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill \\ \hfill a_{n+q-1}\hfill & \hfill a_{n+q-2}\hfill & \hfill \cdots \hfill & \hfill a_{n+2q-2}\hfill \end{array}\mid .$$

(1.4)

The Hankel determinant plays an important role, for instance, in the study of the singularities by Hadamard, see [[18], p. 329], Edrei [19] and in the study of power series with integral coefficients by Polya [[20], p. 323], Cantor [21], and many others.

In this article, we shall determine the rate of growth of the Hankel determinant ${\mathcal{H}}_q\left(n\right)$ for $f\left(z\right)\in {\tilde{\mathcal{B}}}_k\left(\lambda ,\sigma ,\beta ,\gamma \right)$with 0 < β < 2, as n → ∞. This determinant has been considered by several authors. That is, Noor [22] 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 [23] 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

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

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 [17].

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

• (i) there exists a z₁ with |z₁| = r such that for all z, |z| = r

$$\mid z-z_1\mid \mid h_1\left(z\right)\mid \le \frac{2r^2}{1-r^2},$$

   see [25]

(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[26].

2 Hankel determinant problem

Theorem 2.1. Let $f\left(z\right)\in {\tilde{\mathcal{B}}}_k\left(\lambda ,\sigma ,\beta ,\gamma \right)$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

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

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 [1] 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^iθ, 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

$$\begin{array}{c}\mid {\mathrm{\Delta }}_j\left(n,z_1,F\left(z\right)\right)\mid \le \frac{1}{2\pi r^{n+j}}\underset{0}{\overset{2\pi }{\int }}\mid z-z_1{\mid }^j\mid z{\mid }^{\gamma }\mid f\left(z\right){\mid }^{1-\gamma }\left|{\left[\frac{{\left(h_1\left(z\right)/z\right)}^{\left(\frac{k}{4}+\frac{1}{2}\right)}}{{\left(h_2\left(z\right)/z\right)}^{\left(\frac{k}{4}-\frac{1}{2}\right)}}\right]}^{\left(1-\sigma \right)\gamma e^{-i\lambda }\mathsf{\text{cos}}\lambda }\right|\mid p\left(z\right){\mid }^{\beta }d\theta \\ \le \frac{{\left(M\left(r\right)\right)}^{1-\gamma }}{2\pi r^{n+j}}\underset{0}{\overset{2\pi }{\int }}\mid z-z_1{\mid }^j\left|{\left[\frac{{\left(h_1\left(z\right)/z\right)}^{\left(\frac{k}{4}+\frac{1}{2}\right)}}{{\left(h_2\left(z\right)/z\right)}^{\left(\frac{k}{4}-\frac{1}{2}\right)}}\right]}^{\left(1-\sigma \right)\gamma e^{-i\lambda }\mathsf{\text{cos}}\lambda }\right|\mid p\left(z\right){\mid }^{\beta }d\theta .\end{array}$$

(2.3)

Since

$$\begin{array}{cc}\hfill \left|{\left[\frac{{\left(h_1\left(z\right)/z\right)}^{\left(\frac{k}{4}+\frac{1}{2}\right)}}{{\left(h_2\left(z\right)/z\right)}^{\left(\frac{k}{4}-\frac{1}{2}\right)}}\right]}^{\left(1-\sigma \right)\gamma e^{-i\lambda }\mathsf{\text{cos}}\lambda }\right|& \le {\left[\frac{{\mid \left(h_1\left(z\right)/z\right)\mid }^{\left(\frac{k}{4}+\frac{1}{2}\right)}}{{\mid \left(h_2\left(z\right)/z\right)\mid }^{\left(\frac{k}{4}-\frac{1}{2}\right)}}\right]}^{\gamma \left(1-\sigma \right){\mathsf{\text{cos}}}^2\lambda }{e}^{\gamma \left(1-\sigma \right)\gamma \frac{\pi }{2}\frac{\mathsf{\text{sin}}2\lambda }{2}}\hfill \\ =\frac{{\mid \left(h_1\left(z\right)/z\right)\mid }^{\left(\frac{k}{4}+\frac{1}{2}\right)\gamma \left(1-\sigma \right){\mathsf{\text{cos}}}^2\lambda }}{{\mid \left(h_2\left(z\right)/z\right)\mid }^{\left(\frac{k}{4}-\frac{1}{2}\right)\gamma \left(1-\sigma \right){\mathsf{\text{cos}}}^2\lambda }}{c}_1,\mathsf{\text{say}}\hfill \end{array}$$

therefore (2.3) becomes

$$\mid {\mathrm{\Delta }}_j\left(n,z_1,F\left(z\right)\right)\mid \le \frac{c_1{\left(M\left(r\right)\right)}^{1-\gamma }}{2\pi r^{n+j}}\underset{0}{\overset{2\pi }{\int }}{\mid z-z_1\mid }^j\frac{{\mid \left(h_1\left(z\right)/z\right)\mid }^{\left(\frac{k}{4}+\frac{1}{2}\right)\gamma \left(1-\sigma \right){\mathsf{\text{cos}}}^2\lambda }}{{\mid \left(h_2\left(z\right)/z\right)\mid }^{\left(\frac{k}{4}-\frac{1}{2}\right)\gamma \left(1-\sigma \right){\mathsf{\text{cos}}}^2\lambda }}{\mid p\left(z\right)\mid }^{\beta }d\theta .$$

Now using Lemma 1.3, we have

$$\begin{array}{cc}\hfill \mid {\mathrm{\Delta }}_j\left(n,z_1,F\left(z\right)\right)\mid & \le \frac{c_1{\left(M\left(r\right)\right)}^{1-\gamma }}{2\pi r^{n+j}}{\left(\frac{{\left(1+r\right)}^2}{r}\right)}^{\left(\frac{k}{4}-\frac{1}{2}\right)\gamma \left(1-\sigma \right){\mathsf{\text{cos}}}^2\lambda }\hfill \\ \frac{1}{r^{\gamma \left(1-\sigma \right){\mathsf{\text{cos}}}^2\lambda }}{\left(\frac{2r^2}{1-r^2}\right)}^j\underset{0}{\overset{2\pi }{\int }}\mid \left(h_1\left(z\right)\right){\mid }^{\left(\frac{k}{4}+\frac{1}{2}\right)\gamma \left(1-\sigma \right){\mathsf{\text{cos}}}^2\lambda -j}\mid p\left(z\right){\mid }^{\beta }d\theta .\hfill \end{array}$$

The well-known Holder's inequality will give us

$$\begin{array}{cc}\hfill \mid {\mathrm{\Delta }}_j\left(n,z_1,F\left(z\right)\right)\mid & \le \frac{c_1{\left(M\left(r\right)\right)}^{1-\gamma }{2}^{\left(\frac{k}{4}-\frac{1}{2}\right)\gamma \left(1-\sigma \right){\mathsf{\text{cos}}}^2\lambda +j}}{r^{n-j+\left(\frac{k}{4}+\frac{1}{2}\right)\gamma \left(1-\sigma \right){\mathsf{\text{cos}}}^2\lambda }}{\left(\frac{1}{1-r}\right)}^j{\left(\frac{1}{2\pi }\underset{0}{\overset{2\pi }{\int }}{\mid p\left(z\right)\mid }^{2}d\theta \right)}^{\frac{\beta }{2}}\hfill \\ {\left(\frac{1}{2\pi }{\underset{0}{\overset{2\pi }{\int }}\left|\left(h_1\left(z\right)\right)\right|}^{\frac{\left(\frac{k}{2}+1\right)\gamma \left(1-\sigma \right){\mathsf{\text{cos}}}^2\lambda -2j}{2-\beta }}d\theta \right)}^{\frac{2-\beta }{2}}.\hfill \end{array}$$

(2.4)

Also, it is known [15] that, for $p\left(z\right)\in \mathcal{P}$, z ∈ E,

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

(2.5)

Using (2.5) in (2.4), we obtain

$$\begin{array}{cc}\hfill \mid {\mathrm{\Delta }}_j\left(n,z_1,F\left(z\right)\right)\mid & \le \frac{c_1{\left(M\left(r\right)\right)}^{1-\gamma }{2}^{\left(\frac{k}{4}-\frac{1}{2}\right)\gamma \left(1-\sigma \right){\mathsf{\text{cos}}}^2\lambda +j}}{r^{n-j+\left(\frac{k}{4}+\frac{1}{2}\right)\gamma \left(1-\sigma \right){\mathsf{\text{cos}}}^2\lambda }}{\left(\frac{1}{1-r}\right)}^j\hfill \\ {\left(\frac{1+3r^2}{1-r^2}\right)}^{\frac{\beta }{2}}{\left(\frac{1}{2\pi }\underset{0}{\overset{2\pi }{\int }}\left|\left(h_1\left(z\right)\right)\right|\frac{\left(\frac{k}{4}+1\right)\gamma \left(1-\sigma \right){\mathsf{\text{cos}}}^2\lambda -2j}{2-\beta }d\theta \right)}^{\frac{2-\beta }{2}}.\hfill \end{array}$$

Therefore, we can write

$$\begin{array}{c}\mid {\mathrm{\Delta }}_j\left(n,z_1,F\left(z\right)\right)\mid \le \frac{c_1{\left(M\left(r\right)\right)}^{1-\gamma }{2}^{\left(\frac{k}{4}-\frac{1}{2}\right)\gamma \left(1-\sigma \right){\mathsf{\text{cos}}}^2\lambda +\beta +j}}{r^{n-j+\left(\frac{k}{4}+\frac{1}{2}\right)\gamma \left(1-\sigma \right){\mathsf{\text{cos}}}^2\lambda }}{\left(\frac{1}{1-r}\right)}^{j+\frac{\beta }{2}}\\ {\left[r^{\left(\frac{k}{4}+\frac{1}{2}\right)\gamma \left(1-\sigma \right){\mathsf{\text{cos}}}^2\lambda -j}\frac{1}{2\pi }\underset{0}{\overset{2\pi }{\int }}\frac{d\theta }{{\mid 1-r{e}^{i\theta }\mid }^{\frac{\left(k+2\right)\gamma \left(1-\sigma \right){\mathsf{\text{cos}}}^2\lambda -4j}{2-\beta }}}\right]}^{\frac{2-\beta }{2}}.\end{array}$$

Now using a subordination result for starlike functions, we have

$$\begin{array}{cc}\hfill \mid {\mathrm{\Delta }}_j\left(n,z_1,F\left(z\right)\right)\mid & \le \frac{c_2{\left(M\left(r\right)\right)}^{1-\gamma }}{r^n}{\left(\frac{1}{1-r}\right)}^{j+\frac{\beta }{2}}{\left[{\left(\frac{1}{1-r}\right)}^{\frac{\gamma \left(k+2\right)\left(1-\sigma \right){\mathsf{\text{cos}}}^2\lambda -4j}{2-\beta }-1}\right]}^{\frac{2-\beta }{2}},\hfill \\ =\frac{c_2{\left(M\left(r\right)\right)}^{1-\gamma }}{r^n}{\left(\frac{1}{1-r}\right)}^{\gamma \left(\frac{k}{2}+1\right)\left(1-\sigma \right){\mathsf{\text{cos}}}^2\lambda +\beta -1-j},\hfill \end{array}$$

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 $z_1=\left(\frac{n}{n+1}\right)e^{i{\theta }_n},\left(n\to \infty \right),r=1-\frac{1}{n}$, we have for $k\ge \frac{4j-\beta +2}{\left(1-\sigma \right)\gamma {\mathsf{\text{cos}}}^2\lambda }-2$,

$$\mid {\mathrm{\Delta }}_j\left(n,e^{i{\theta }_n},f\left(z\right)\right)\mid =\mathsf{\text{O}}\left(1\right){\left(M\left(r\right)\right)}^{1-\gamma }{n}^{\gamma \left(\frac{k}{2}+1\right)\left(1-\sigma \right){\mathsf{\text{cos}}}^2\lambda +\beta -j-2},$$

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

$$H_1\left(n\right)=a_n=\mathsf{\text{O}}\left(1\right){\left(M\left(r\right)\right)}^{1-\gamma }{n}^{\gamma \left(\frac{k}{2}+1\right)\left(1-\sigma \right){\mathsf{\text{cos}}}^2\lambda +\beta -2}.$$

For q ≥ 2, we use similar argument due to Noonan and Thomas [17] together with Lemma 1.1 to have

$${\mathcal{H}}_q\left(n\right)=\mathsf{\text{O}}\left(1\right){\left(M\left(r\right)\right)}^{1-\gamma }{n}^{\left[\gamma \left(\frac{k}{2}+1\right)\left(1-\sigma \right){\mathsf{\text{cos}}}^2\lambda +\beta -1\right]q-q^2},k\ge \frac{8\left(q-1\right)}{\left(1-\sigma \right)\gamma {\mathsf{\text{cos}}}^2\lambda }-2$$

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. [14].

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. [14] and

$${\mathcal{H}}_q\left(n\right)=\mathsf{\text{O}}\left(1\right){\left(M\left(r\right)\right)}^{1-\gamma }\left\{\begin{array}{c}{n}^{\gamma \left(\frac{k}{2}+1\right)-1},q=1\\ n^{\gamma \left(\frac{k}{2}+1\right)q-q^2},q\ge 2,k\ge \frac{8\left(q-1\right)}{\gamma }-2\end{array}\right.$$

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

Corollary 2.2. For $\lambda =0,\gamma =1,f\left(z\right)\in {\tilde{\mathcal{T}}}_k\left(\beta ,\sigma \right)$ and

$${\mathcal{H}}_q\left(n\right)=\mathsf{\text{O}}\left(1\right)\left\{\begin{array}{c}{n}^{\left(1-\sigma \right)\left(\frac{k}{2}+1\right)+\beta -2},q=1\\ n^{\left[\left(1-\sigma \right)\left(\frac{k}{2}+1\right)+\beta -1\right]q-q^2},q\ge 2,k\ge \frac{8\left(q-1\right)}{\left(1-\sigma \right)}-2\end{array}\right.$$

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