Coefficient estimates for the class of q-starlike functions

Lemma 2.1. [31] For a function $\omega(z) = b_1 z + b_2 z^2 + \cdots \in \mathcal{B}_0$ , if $b_1 > 0$ , then there are complex numbers x, y such that (9) $$b_2 = x(1 - b_1^2), \quad b_3 = (1 - b_1^2)[(1 - |x|^2)y - b_1x^2],$$ and $|x| \le 1, |y| \le 1.$ Remark 1. When $b_1$ is a real number, $p_1$ is also real. Furthermore, if we set x = -1 in (7) (i.e., $p_2 = p_1^2 - 2$ ), then by a lemma due to Li and Sugawa (Lemma 2.3 in [32]), we obtain $p(z) = (1 - z^2)/(1 - p_1 z + z^2)$ . Consequently, from (5) it follows that either $\omega(z) = z(2z - p_1)/(p_1 z - 2)$ or $\omega(z) = z(z - b_1)/(b_1 z - 1)$ . Remark 2. Observe that when $b_1 = 1$ , we have $b_2 = 0$ and $b_3 = 0$ . Schwarz lemma states that the only function in $\mathcal{B}_0$ with $b_1 = 1$ is $\omega(z) = z$ . This paper concerns two fundamental functionals for class $\mathcal{B}_0$ , with the primary result established by Prokhorov and Szynal [33].

Lemma 2.2. [33] If an analytic function $\omega(z) = b_1 z + b_2 z^2 + \cdots \in \mathcal{B}_0$ , then for all real numbers $\mu, \nu$ , the following sharp estimate holds: $$|b_3 + \mu b_1 b_2 + \nu b_1^3| \le |\nu|$$ for $(\mu, \nu) \in D_1$ where $$D_1 = \left\{ (\mu, \nu) : |\mu| \ge \frac{1}{2}, \ \nu \le -\frac{2}{3} (|\mu| + 1) \ or \ |\mu| \ge 4, \ \nu \ge \frac{2}{3} (|\mu| - 1) \right\}.$$ The second functional is (10) $$Y(a,b,c) = \max_{z \in \mathbb{D}} (|a+bz+cz^2| + 1 - |z|^2),$$ which is central to our proofs and connects to Ohno-Sugawa's [34] findings.

Lemma 2.3. [34] Let $a, b, c \in \mathbb{R}$ , if $ac \geq 0$ , then $$Y(a,b,c) = \begin{cases} |a| + |b| + |c|, & (|b| \ge 2(1 - |c|)), \\ 1 + |a| + \frac{b^2}{4(1 - |c|)}, & (|b| < 2(1 - |c|)). \end{cases}$$ As a special case, we have the following lemma.

Lemma 2.4. For $q \in (0,1)$ and $b_1 \in (0,1)$ , if a, b and c are given by (11) $$a = \frac{-(2+q)b_1^3}{(1-b_1^2)(1+q)^2}, \quad b = \frac{2b_1}{(1+q)^2}, \quad c = -b_1 - \frac{(1+q+q^2)(1-b_1^2)(1+q)^2}{b_1},$$ then $$Y(a,b,c) = \frac{(1-q)b_1}{(1+q)(1-b_1^2)} + \frac{1+q+q^2}{b_1(1-b_1^2)(1+q)^2}.$$

Theorem 3.1. For $$f(z) = z + \sum_{n=2}^{\infty} a_n z^n \in S_q^*$$ , $q \in (0,1)$ , we have $$|a_2| \le \frac{2}{q}, \qquad |a_3| \le \frac{4+2q}{q^2(1+q)}, \quad |a_4| \le \frac{2(4+4q+3q^2+q^3)}{q^3(1+q+q^2)(1+q)}.$$ These estimates are sharp, with extremal function (12) $$f(z) = z \prod_{k=0}^{\infty} \frac{q - q^{k+1}z}{q - q^k(2 - q)z}.$$

Theorem 4.1. If a function $f(z) = z + \sum_{n=2}^{\infty} a_n z^n \in S_q^*$ and $q \in (0,1)$ , then $$|a_2 a_3 - a_4| \le \frac{2(q+2)}{q^2(1+q+q^2)}.$$ This estimate is sharp.

Theorem 4.2. If a function $f(z) = z + \sum_{n=2}^{\infty} a_n z^n \in S_q^*$ and $q \in (0,1)$ , then $$|H_1^{(2)}| = |a_3 - a_2^2| \le \frac{2}{q(q+1)},$$ $$|H_2^{(2)}| = |a_2 a_4 - a_3^2| \le \begin{cases} \frac{4}{q^2 (1+q)^2}, & (a_2 = 0), \\ \frac{4(2+q)}{q^2 (1+q)^2 (1+q+q^2)}, & (a_2 \ne 0). \end{cases}$$ These estimates are sharp.

Theorem 4.3. If a function $$f(z) = z + \sum_{n=2}^{\infty} a_n z^n \in S_q^*, \ q \in (0,1), \ then$$ $$|T_1^{(2)}| = |a_1^2 - a_2^2| \le 1 + \frac{4}{q^2},$$ $$|T_2^{(2)}| = |a_2^2 - a_3^2| \le \frac{4}{q^4} + \frac{4(2+q)^2}{q^4(1+q)^2}$$ $$|T_3^{(2)}| = |a_3^2 - a_4^2| \le \frac{4(2+q)^2}{q^4(1+q)^2} + \frac{4(4+4q+3q^2+q^3)^2}{q^6(1+q+q^2)^2(1+q)^2}.$$ These estimates are sharp.

Theorem 4.4. If a function $f(z) = z + \sum_{n=2}^{\infty} a_n z^n \in S_q^*, \ q \in (0,1), \ then$ $$|T_1^{(3)}| \le 1 + \frac{8}{q^2} + \frac{4(3q^2 + 8q + 4)}{q^4(1+q)^2}.$$ This estimate is sharp.

Theorem 4.5. If a function $$f(z) = z + \sum_{n=2}^{\infty} a_n z^n \in S_q^*$$ , $q \in (0,1)$ , then $$|T_2^{(3)}| \le \frac{8(4+4q+3q^2+2q^3+q^4)(4+4q+4q^2+3q^3+2q^4+q^5)}{a^7(1+q)^3(1+q+q^2)}$$ when $a_2 = 0$ , and $$|T_2^{(3)}| \le \frac{8(4+4q+4q^2+3q^3+2q^4+q^5)(4+8q+12q^2+9q^3+5q^4+3q^5+q^6)}{q^7(1+q)^3(1+q+q^2)^2}$$ for $a_2 \neq 0$ . These estimates are sharp.

Theorem 5.1. If we assume $|\zeta| \leq 1$ , $\alpha \in [0,1)$ , and $f(z) = z + \sum_{n=2}^{\infty} a_n z^n \in \mathcal{S}_{\zeta}^*(\alpha)$ , then $$|a_n|^2 \le \frac{\sum_{k=1}^{n-1} \left\{ |(1-2\alpha) + [k]_{\zeta}|^2 - |[k]_{\zeta} - 1|^2 \right\} |a_k|^2}{|[n]_{\zeta} - 1|^2}.$$

Theorem 5.2. For positive integers $k \in \mathbb{N}^+$ , let $P(k) = |(1-2\alpha)+[k]_{\zeta}|^2 - |[k]_{\zeta}-1|^2$ and $Q(k) = \frac{(1-2\alpha)+[k-1]_{\zeta}}{[k]_{\zeta}-1}$ . If $\zeta$ satisfies $\text{Re }([k]_{\zeta}) > \alpha$ for $\alpha \in (0,1)$ , then (34) $$\frac{1}{\left|[n]_{\zeta} - 1\right|^{2}} \sum_{k=1}^{n-1} P(k) \left|a_{k}\right|^{2} \leq \prod_{k=2}^{n} \left|Q(k)\right|^{2}.$$

Theorem 5.3. For $f \in \mathcal{S}^*_{\zeta}(\alpha)$ , if $\zeta$ satisfies $\text{Re }([k]_{\zeta}) > \alpha$ for $\alpha \in [0,1)$ , then we have (36) $$|a_n| \le \prod_{k=2}^n \frac{(1-2\alpha) + [k-1]_{\zeta}}{[k]_{\zeta} - 1}.$$ When $\alpha = 0$ and $\zeta$ is real (Re $[k]_{\zeta} > \alpha$ ), Theorem 5.3 implies the following: Corollary 5.4. For $f(z) = z + \sum_{n=2}^{\infty} a_n z^n \in S_q^*$ with $q \in (0,1)$ , we have $$|a_n| \le \prod_{k=2}^n \frac{1 + [k-1]_{\zeta}}{[k]_{\zeta} - 1}.$$ This estimate is sharp, and the extremal function is given by $$f(z) = z \prod_{k=0}^{\infty} \frac{\zeta - \zeta^{k+1} z}{\zeta - \zeta^k (2 - \zeta) z}.$$ Remark 10. It is a result of Agrawal proved in [8] (Theorem 3.3).

Definition 1.1. [3, 4] Let $\zeta \in \mathbb{C}$ with $|\zeta| \leq 1$ . An analytic function $f \in \mathcal{A}$ is said to be $\zeta$ -starlike of order $\alpha$ if it satisfies the condition (4) $$\operatorname{Re} \frac{zD_{\zeta}f(z)}{f(z)} > \alpha, \quad z \in \mathbb{D},$$ where $\alpha \in [0,1)$ . The class of all such functions is denoted by $S_{\zeta}^{*}(\alpha)$ . It is worth noting that if we take $\zeta = q$ and $q \in (0,1)$ , $D_{\zeta}$ reduces to the q-derivative [5], which is defined by $$D_q f(z) = \begin{cases} \frac{f(z) - f(qz)}{(1 - q)z}, & z \neq 0, \\ f'(0), & z = 0. \end{cases}$$ It is clear that $\lim_{q\to 1^-} D_q f(z) = f'(z)$ . By using the q-derivative, various families of q-extensions of starlike functions have emerged. For example, Ismail et al. [6] introduced a class of generalized starlike functions: $PS_q^* = \left\{ f \in \mathcal{A} : \left| \frac{zD_q f(z)}{f(z)} - \frac{1}{1-q} \right| \le \frac{1}{1-q} \right\}$ . Agrawal [8] investigated the class of q-starlike functions of order $\alpha$ , which constitutes Agrawal [8] investigated the class of q-starlike functions of order $\alpha$ , which constitutes a particular instance of $\zeta$ -starlike functions when $\zeta = q$ . Throughout this paper, we will use the simplified notation $S_q$ instead of $S_q^(0)$ . The Hankel and Toeplitz determinants (defined in Section 4) play an important role in the study of singularities, power series with integral coefficients, and many other areas [9-11]. The upper bounds of them for starlike functions have been thoroughly examined by various researchers [12-24]. Related studies on q-starlike functions have appeared separately in [8, 25-27]. In this paper, we systematically investigates the coefficient properties and Hankel determinants of analytic function family $S_{\zeta}^(\alpha)$ . In section 3, we focus on the Bieberbach-type problem to the family $S_q$ (where q is a real number). We provide upper bounds of $|a_n|$ for $f \in S_q$ . The precise bounds of coefficients $|a_2|, |a_3|, |a_4|$ for $f \in S_q$ are provided. In section 4, we examine the Hankel and Toeplitz determinants for the coefficients of functions in the family $S_q$ , not only significantly improving Zaprawa's results [S. Agrawal, Indagationes Mathematicae, 2021, Theorem 3.4] on the upper bounds of second-order Hankel determinants but also extending the research methodology to the estimation problem of higher-order determinants. Notably, in section 5, we expand the research perspective to the complex domain, conducting an in-depth exploration of the Bieberbach-type problem to family $S_{\zeta}^(\alpha)$ (with parameter $\zeta \in \mathbb{C}$ and $|\zeta| \leq 1$ ) and obtaining a series of new findings with theoretical significance.