Hankel determinant problem of a subclass of

Lemma 1.1. Let $f(z) \in A$ . Let the qth Hankel determinant of f(z) for $q \ge 1$ , $n \ge 1$ be defined by (1.4). Then, writting $\Delta_i(n) = \Delta_i(n, z_1, f(z))$ , we have $$\mathcal{H}_{q}(n) = \begin{vmatrix} \Delta_{2q-2}(n) & \Delta_{2q-3}(n+1) & \cdots & \Delta_{q-1}(n+q-1) \\ \Delta_{2q-3}(n+1) & \Delta_{2q-4}(n+2) & \cdots & \Delta_{q-2}(n+q-2) \\ \vdots & \vdots & \vdots & \vdots \\ \Delta_{q-1}(n+q-1) & \Delta_{q-2}(n+q-2) & \cdots & \Delta_{q}(n+2q-2) \end{vmatrix},$$ where with $\Delta_0(n) = a_n$ , we define for $j \ge 1$ , $$\Delta_{i}(n, z_{1}, f(z)) = \Delta_{i-1}(n, z_{1}, f(z)) - \Delta_{i-1}(n+1, z_{1}, f(z)). \tag{1.5}$$

Lemma 1.2. With $z_1 = \frac{n}{n+1} \gamma$ and $\nu \ge 0$ any integer, $$\Delta_{j}(n+v,z_{1},zf'(z)) = \sum_{m=0}^{j} {j \choose m} \frac{\gamma^{m}(v-(m-1)n)}{(n+1)^{m}} \Delta_{j-m}(n+m+v,f(z)).$$ Lemmas 1.1 and 1.2 are due to Noonan and Thomas [17].

Lemma 1.3. Let $h_1(z)$ be starlike univalent function in $\mathscr{E}$ . Then (i) there exists a $z_1$ with $|z_1| = r$ such that for all z, |z| = r $$|z-z_1||h_1(z)| \leq \frac{2r^2}{1-r^2}$$ (ii) $$\frac{r}{\left(1+r\right)^{2}}\leq \mid h_{1}(z)\mid \leq \frac{r}{\left(1+r\right)^{2}},$$ see[26].

Theorem 2.1. Let $f(z) \in \tilde{\mathcal{B}}_k(\lambda, \sigma, \beta, \gamma)$ with $0 < \beta < 2$ and let the qth Hankel determinant $\mathcal{H}_q(n)$ of f(z) be defined as in (1.4). Then $$\mathcal{H}_{q}(n) = O(1) (M(r))^{1-\gamma} \begin{cases} r \left(\frac{k}{2}+1\right)^{(1-\sigma)\cos^{2}\lambda+\beta-2}, & q = 1\\ r \left\{r \left(\frac{k}{2}+1\right)^{(1-\sigma)\cos^{2}\lambda+\beta-1}\right\}^{q-q^{2}}, & q \geq 2, k \geq \frac{8(q-1)}{(1-\sigma)\cos^{2}\lambda} - 2, \end{cases}$$ where $k > \frac{4j+2-\beta}{(1-\sigma)\gamma\cos^2\lambda} - 2$ and O(1) is a constant depending on k, $\lambda$ , $\beta$ , $\sigma$ , $\gamma$ , and j only. Proof. It is well known [1] that for starlike functions $h_1(z)$ and $h_2(z)$ $$f_1'(z) = \left[ \frac{(h_1(z)/z)^{\left(\frac{k}{4} + \frac{1}{2}\right)}}{(h_2(z)/z)^{\left(\frac{k}{4} - \frac{1}{2}\right)}} \right]^{(1-\sigma)\gamma e^{-i\lambda}\cos\lambda}.$$ $$(2.1)$$ Using (2.1) in (1.3), we have $$zf'(z) = z^{\gamma} (f(z))^{1-\gamma} \left[ \frac{(h_1(z)/z) \left(\frac{k}{4} + \frac{1}{2}\right)}{(h_2(z)/z) \left(\frac{k}{4} - \frac{1}{2}\right)} \right]^{(1-\sigma)\gamma e^{-i\lambda} \cos \lambda}$$ $$p^{\beta}(z).$$ $$(2.2)$$ where $p(z) \in \mathcal{P}$ . Let F(z) = zf'(z). Then for $j \ge 1$ , $z_1$ any non-zero complex and $z = re^{i\theta}$ , consider $\Delta_j(n, z_1, F(z))$ as defined by (1.5). Then $$\mid \Delta_j(n,z_1,F(z))\mid = \frac{1}{2\pi r^{n+j}} \left| \int_0^{2\pi} (z-z_1)^j F(z) e^{i(n+j)\theta} d\theta \right|,$$ and by using (2.2), we have $$| \Delta_{j}(n, z_{1}, F(z)) | \leq \frac{1}{2\pi r^{n+j}} \int_{0}^{2\pi} |z - z_{1}|^{j} |z|^{\gamma} |f(z)|^{1-\gamma} \left[ \frac{(h_{1}(z)/z)^{\left(\frac{k}{4} + \frac{1}{2}\right)}}{(h_{2}(z)/z)^{\left(\frac{k}{4} - \frac{1}{2}\right)}} \right]^{(1-\sigma)\gamma e^{-i\lambda} \cos \lambda} |p(z)|^{\beta} d\theta$$ $$\leq \frac{(M(r))^{1-\gamma}}{2\pi r^{n+j}} \int_{0}^{2\pi} |z - z_{1}|^{j} \left[ \frac{(h_{1}(z)/z)^{\left(\frac{k}{4} + \frac{1}{2}\right)}}{(h_{2}(z)/z)^{\left(\frac{k}{4} - \frac{1}{2}\right)}} \right]^{(1-\sigma)\gamma e^{-i\lambda} \cos \lambda} |p(z)|^{\beta} d\theta.$$ $$(2.3)$$ Since $$\left| \left[ \frac{(h_1(z)/z)^{\left(\frac{k}{4} + \frac{1}{2}\right)}}{(h_2(z)/z)^{\left(\frac{k}{4} - \frac{1}{2}\right)}} \right]^{(1-\sigma)\gamma e^{-i\lambda}\cos\lambda} \right| \leq \left[ \frac{\left| (h_1(z)/z) \right|^{\left(\frac{k}{4} + \frac{1}{2}\right)}}{\left| (h_2(z)/z) \right|^{\left(\frac{k}{4} - \frac{1}{2}\right)}} \right]^{\gamma(1-\sigma)\cos^2\lambda} e^{\gamma(1-\sigma)\gamma\frac{\pi}{2}\frac{\sin 2\lambda}{2}}$$ $$= \frac{\left| (h_1(z)/z) \right|^{\left(\frac{k}{4} + \frac{1}{2}\right)\gamma(1-\sigma)\cos^2\lambda}}{\left| (h_2(z)/z) \right|^{\left(\frac{k}{4} - \frac{1}{2}\right)\gamma(1-\sigma)\cos^2\lambda}} c_1, \text{ say}$$ therefore (2.3) becomes $$|\Delta_{j}(n, z_{1}, F(z))| \leq \frac{c_{1}(M(r))^{1-\gamma}}{2\pi r^{n+j}} \int_{0}^{2\pi} |z-z_{1}|^{j} \frac{|(h_{1}(z)/z)|^{\left(\frac{k}{4}+\frac{1}{2}\right)\gamma(1-\sigma)\cos^{2}\lambda}}{|(h_{2}(z)/z)|^{\left(\frac{k}{4}-\frac{1}{2}\right)\gamma(1-\sigma)\cos^{2}\lambda}} |p(z)|^{\beta} d\theta.$$ Now using Lemma 1.3, we have $$| \Delta_{j}(n, z_{1}, F(z)) | \leq \frac{c_{1}(M(r))^{1-\gamma}}{2\pi r^{n+j}} \left( \frac{(1+r)^{2}}{r} \right)^{\left(\frac{k}{4} - \frac{1}{2}\right)\gamma(1-\sigma)\cos^{2}\lambda}$$ $$\frac{1}{r^{\gamma(1-\sigma)\cos^{2}\lambda}} \left( \frac{2r^{2}}{1-r^{2}} \right)^{j} \int_{0}^{2\pi} | (h_{1}(z))|^{\left(\frac{k}{4} + \frac{1}{2}\right)\gamma(1-\sigma)\cos^{2}\lambda - j} | p(z)|^{\beta} d\theta.$$ The well-known Holder's inequality will give us $$|\Delta_{j}(n, z_{1}, F(z))| \leq \frac{c_{1}(M(r))^{1-\gamma} 2^{\left(\frac{k}{4} - \frac{1}{2}\right)\gamma(1-\sigma)\cos^{2}\lambda + j}}{r^{n-j+\left(\frac{k}{4} + \frac{1}{2}\right)\gamma(1-\sigma)\cos^{2}\lambda}} \left(\frac{1}{1-r}\right)^{j} \left(\frac{1}{2\pi} \int_{0}^{2\pi} |p(z)|^{2} d\theta\right)^{\frac{\beta}{2}}$$ $$\left(\frac{1}{2\pi} \int_{0}^{2\pi} |h_{1}(z)| + \frac{\left(\frac{k}{2} + 1\right)\gamma(1-\sigma)\cos^{2}\lambda - 2j}{2-\beta}\right)^{\frac{2-\beta}{2}} d\theta$$ $$d\theta$$ $$(2.4)$$ Also, it is known [15] that, for $p(z) \in \mathcal{P}$ , $z \in E$ , $$\frac{1}{2\pi} \int_{0}^{2\pi} |p(z)|^2 d\theta \le \frac{1+3r^2}{1-r^2}.$$ (2.5) Using (2.5) in (2.4), we obtain $$|\Delta_{j}(n, z_{1}, F(z))| \leq \frac{c_{1}(M(r))^{1-\gamma} 2^{\left(\frac{k}{4}-\frac{1}{2}\right)\gamma(1-\sigma)\cos^{2}\lambda+j}}{r^{n-j+\left(\frac{k}{4}+\frac{1}{2}\right)\gamma(1-\sigma)\cos^{2}\lambda}} \left(\frac{1}{1-r}\right)^{j}$$ $$\left(\frac{1+3r^{2}}{1-r^{2}}\right)^{\frac{\beta}{2}} \left(\frac{1}{2\pi} \int_{0}^{2\pi} |(h_{1}(z))| \frac{\left(\frac{k}{4}+1\right)\gamma(1-\sigma)\cos^{2}\lambda-2j}{2-\beta} d\theta\right)^{\frac{2-\beta}{2}}.$$ Therefore, we can write $$|\Delta_{j}(n, z_{1}, F(z))| \leq \frac{c_{1}(M(r))^{1-\gamma} 2^{\left(\frac{k}{4} - \frac{1}{2}\right)\gamma(1-\sigma)\cos^{2}\lambda + \beta + j}}{r^{n-j+\left(\frac{k}{4} + \frac{1}{2}\right)\gamma(1-\sigma)\cos^{2}\lambda}} \left(\frac{1}{1-r}\right)^{j+\frac{\beta}{2}}$$ $$\left[r^{\left(\frac{k}{4} + \frac{1}{2}\right)\gamma(1-\sigma)\cos^{2}\lambda - j} \frac{1}{2\pi} \int_{0}^{2\pi} \frac{d\theta}{\left(\frac{k+2)\gamma(1-\sigma)\cos^{2}\lambda - 4j}{2-\beta}\right]^{\frac{2-\beta}{2}}}\right]^{\frac{2-\beta}{2}}.$$ Now using a subordination result for starlike functions, we have $$|\Delta_{j}(n, z_{1}, F(z))| \leq \frac{c_{2}(M(r))^{1-\gamma}}{r^{n}} \left(\frac{1}{1-r}\right)^{j+\frac{\beta}{2}} \left[ \left(\frac{1}{1-r}\right)^{\frac{\gamma(k+2)(1-\sigma)\cos^{2}\lambda-4j}{2-\beta}} \right]^{\frac{2-\beta}{2}},$$ $$= \frac{c_{2}(M(r))^{1-\gamma}}{r^{n}} \left(\frac{1}{1-r}\right)^{\frac{\gamma(k+2)(1-\sigma)\cos^{2}\lambda+\beta-1-j}{2-\beta}},$$ where $c_2$ is a constant depending on k, $\lambda$ , $\beta$ , $\sigma$ , $\gamma$ , j only and $\gamma(k+2)$ $(1-\sigma)$ $\cos^2 \lambda - 4j > 2-\beta$ . Applying Lemma 1.2 and putting $z_1 = \left(\frac{n}{n+1}\right)e^{i\theta_n}$ , $(n \to \infty)$ , $r = 1 - \frac{1}{n}$ , we have for $|\Delta_j(n, e^{i\theta_n}, f(z))| = O(1)(M(r))^{1-\gamma}n^{\gamma\left(\frac{k}{2}+1\right)(1-\sigma)\cos^2\lambda + \beta - j - 2}$ , $$\mid \Delta_j \left(n, e^{i\theta_n}, f(z)\right) \mid = \mathrm{O}(1) \left(M(r)\right)^{1-\gamma} n^{\gamma \left(\frac{k}{2}+1\right) (1-\sigma) \cos^2 \lambda + \beta - j - 2},$$ where O(1) is a constant depending on k, $\lambda$ , $\beta$ , $\sigma$ , $\gamma$ , and j only. We now estimate the rate of growth of $\mathcal{H}_a(n)$ . For $$q = 1$$ , $\mathcal{H}_q(n) = a_n = \Delta_0(n)$ and $$H_1(n) = a_n = O(1) (M(r))^{1-\gamma} n^{\gamma (\frac{k}{2}+1)(1-\sigma)\cos^2\lambda + \beta - 2}$$ For $q \ge 2$ , we use similar argument due to Noonan and Thomas [17] together with Lemma 1.1 to have $$\mathcal{H}_{q}(n) = O(1) (M(r))^{1-\gamma} n^{\left[\gamma\left(\frac{k}{2}+1\right)(1-\sigma)\cos^{2}\lambda+\beta-1\right]q-q^{2}}, \ k \geq \frac{8(q-1)}{(1-\sigma)\gamma\cos^{2}\lambda} - 2$$ and O(1) depends only on k, $\lambda$ , $\beta$ , $\sigma$ , $\gamma$ , and j. For choosing different values of $\lambda$ , $\beta$ , $\sigma$ , and $\gamma$ in Theorem 2.1, we obtain the following results discussed by Noor [10,11] and Noor et al. [14].

Corollary 2.1. For $\lambda = 0$ , $\beta = 1$ , $\sigma = 0$ , $f(z) \in \mathcal{B}_k(\gamma)$ , where the class $\mathcal{B}_k(\gamma)$ was introduced by Noor et al. [14] and $$\mathcal{H}_{q}(n) = O(1)(M(r))^{1-\gamma} \begin{cases} n^{\gamma \left(\frac{k}{2}+1\right)-1}, & q = 1\\ n^{\gamma \left(\frac{k}{2}+1\right)q-q^{2}}, & q \geq 2, & k \geq \frac{8(q-1)}{\gamma} - 2 \end{cases}$$ where O(1) is a constant depending only on k and $\gamma$ .

Corollary 2.2. For $\lambda = 0$ , $\gamma = 1$ , $f(z) \in \tilde{\mathcal{T}}_k(\beta, \sigma)$ and $$\mathcal{H}_{q}(n) = O(1) \begin{cases} n^{(1-\sigma)\left(\frac{k}{2}+1\right)+\beta-2}, & q = 1\\ n^{\left[(1-\sigma)\left(\frac{k}{2}+1\right)+\beta-1\right]q-q^{2}}, & q \geq 2, & k \geq \frac{8(q-1)}{(1-\sigma)}-2 \end{cases}$$ where O(1) is a constant depends on k, $\sigma$ , and $\beta$ only.

Definition 1.1. Let $f(z) \in \mathcal{A}$ in E. Then $f(z) \in \tilde{\mathcal{B}}_k(\lambda, \sigma, \beta, \gamma)$ , if for $k \geq 2$ , $0 \leq \beta \leq 1$ , $0 \leq \gamma \leq 1$ , $\lambda$ is real with $|\lambda| < \frac{\pi}{2}$ there exists a function $f_1(z) \in \mathcal{V}_k^{\lambda}(\sigma)$ , $0 \leq \sigma < 1$ , such that $$\left|\arg\left\{\frac{z^{1-\gamma}f'(z)}{f(z)}\left(\frac{f(z)}{f_1'(z)}\right)^{\gamma}\right\}\right| \leq \frac{\beta\pi}{2}, \ z \in E.$$  By giving specific values to the parameters k, $\sigma$ , $\lambda$ , $\beta$ , and $\gamma$ in $\tilde{\mathcal{B}}_k(\lambda, \sigma, \beta, \gamma)$ , we obtain many important subclasses studied by various authors in earlier articles, see [10-16]. Using (1.1) and (1.2), we have $$zf'(z) = z^{\gamma} (f(z))^{1-\gamma} (f'_1(z))^{\gamma} p^{\beta}(z), \tag{1.3}$$ where $f_1(z) \in \mathcal{V}_k^{\lambda}(\sigma)$ 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 $k \ge 2$ , $0 \le \beta \le 1$ , $0 < \gamma \le 1$ , $\lambda$ is real with $|\lambda| < \frac{\pi}{2}$ , $0 \le \sigma < 1$ . In [17], the qth Hankel determinant $\mathcal{H}_q(n)$ , $q \ge 1$ , $n \ge 1$ , for a function $f(z) \in \mathcal{A}$ is stated by Noonan and Thomas as:

Definition 1.2. Let $f(z) \in A$ . Then the qth Hankel determinant of f(z) is defined for $q \ge 1$ , $n \ge 1$ by $$H_{q}(n) = \begin{vmatrix} a_{n} & a_{n+1} & \cdots & a_{n+q-1} \\ a_{n+1} & a_{n+2} & \cdots & a_{n+q-2} \\ \vdots & \vdots & \vdots & \vdots \\ a_{n+q-1} & a_{n+q-2} & \cdots & a_{n+2q-2} \end{vmatrix}.$$ $$(1.4)$$ 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 $\mathcal{H}_q(n)$ for $f(z) \in \tilde{\mathcal{B}}_k(\lambda, \sigma, \beta, \gamma)$ with $0 < \beta < 2$ , as $n \to \infty$ . This determinant has been considered by several authors. That is, Noor [22] determined the rate of growth of $\mathcal{H}_q(n)$ as $n \to \infty$ 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].