Sharp Coefficient Bounds for certain q-Starlike Functions

Theorem 1.1. Let $f \in \mathcal{S}_{\xi}^*$ . Then - (1) Subordination results: $\frac{zf'(z)}{f(z)} \prec \frac{z\tilde{f}'(z)}{\tilde{f}(z)}$ and $\frac{f(z)}{z} \prec \frac{\tilde{f}(z)}{z}$ . - (2) Growth theorem: For |z| = r < 1, $-\tilde{f}(-r) \le |f(z)| \le \tilde{f}(r)$ . - (3) Distortion theorem: For |z| = r < 1, $-|1 M(r)| \frac{\tilde{f}(-r)}{r} \le |f'(z)| \le |1 + M(r)| \frac{\tilde{f}(r)}{r}$ , where $M(r) := \max_{|z|=r} \left| \frac{\sin(z)}{(1-z)} \right|$ . - (4) Rotation theorem: For |z| = r < 1, $\left| \arg \frac{f(z)}{z} \right| \le \max_{|z|=r} \arg \frac{\tilde{f}(z)}{z}$ . - (5) Covering theorem: The function f is either a rotation of $\tilde{f}$ , or its image contains the disk $\{w \in \mathbb{C} : |w| < -\tilde{f}(-1)\}$ , where $\tilde{f}(-1) = \lim_{r \to 1^-} \tilde{f}(-r)$ . In an analogous manner, the above results can be extended to the class $\mathcal{S}_{\xi_q}^*$ . A function f belongs to $\mathcal{S}_{\xi_g}^*$ if and only if there exists a Schwarz function $w(z) \in \mathcal{B}_0$ such that $$\frac{zd_q f(z)}{f(z)} = \xi_q(w(z)).$$ This representation yields the integral form $$f(z) = z \exp\left(\int_0^z \frac{\xi_q(w(t)) - \lambda_q}{t} d_q t\right),\,$$ where $\lambda_q = \frac{\ln q}{q-1}$ and $\lim_{q \to 1^-} \lambda_q = 1$ . Using the Jackson integral definition <span id="page-3-0"></span> $$\int_0^z h(t)d_q t = (1 - q)z \sum_{k=0}^\infty q^k h(q^k z),$$ we obtain the explicit series representation $$\int_0^z \frac{\xi_q(w(t)) - \lambda_q}{t} d_q t = (1 - q) \sum_{k=0}^\infty \left( \xi_q(w(q^k z)) - \lambda_q \right),$$ provided the series converges for the given $\xi_q$ and q. The extremal function for the class $\mathcal{S}_{\xi_a}^*$ , corresponding to w(z) = z, is given by $$\tilde{f}_q(z) = z \exp\left(\int_0^z \frac{\xi_q(t) - \lambda_q}{t} d_q t\right)$$ $$= z \exp\left(\int_0^z \frac{\sin(qt) + q(1 - qt)\left(1 + \frac{\ln q}{1 - q}\right)}{qt(1 - qt)} d_q t\right) \in \mathcal{S}_{\xi_q}^*.$$ (1.9) Its classical counterpart for $q \to 1^-$ is <span id="page-3-1"></span> $$\tilde{f}(z) = z \exp\left(\int_0^z \frac{\xi(t) - 1}{t} dt\right) = z \exp\left(\int_0^z \frac{\sin t}{t(1 - t)} dt\right) \in \mathcal{S}_{\xi}^*. \tag{1.10}$$ The extremal function $\tilde{f}_q$ , defined explicitly in equation (1.9), admits an alternative characterization through a convolution equation. Specifically, it is the unique analytic function (normalized by $\tilde{f}_q(0) = 0$ and $\tilde{f}'_q(0) = 1$ ) satisfying the functional relation: $$\tilde{f}_q(z) * \frac{z}{(1-qz)(1-z)} = \tilde{f}_q(z) \cdot \xi_q(z).$$ A Hankel matrix is a square matrix that is symmetric about its principal diagonal, capture nonlinear structural behaviour supporting stable simulations. For functions $f \in \mathcal{S}$ of the form (1.1), Pommerenke [19] defined the sth Hankel determinant as <span id="page-4-1"></span> $$H_{s,n}(f) = \begin{vmatrix} a_n & a_{n+1} & \cdots & a_{n+s-1} \\ a_{n+1} & a_{n+2} & \cdots & a_{n+s} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n+s-1} & a_{n+s} & \cdots & a_{n+2s-2} \end{vmatrix},$$ (1.11) where $n, s \in \mathbb{N}$ and $a_1 = 1$ . Establishing sharp upper bounds for Hankel determinants remains a central problem in geometric function theory. Ye and Lim [24] demonstrated that any $n \times n$ matrix over $\mathbb{C}$ can be expressed as a product of Toeplitz or Hankel matrices. Toeplitz matrices are characterized by constant entries along each diagonal and find extensive applications in quantum physics, image processing, integral equations, and signal processing. The Toeplitz determinant for $f \in \mathcal{S}$ is defined as <span id="page-4-0"></span> $$T_{s,n}(f) = \begin{vmatrix} a_n & a_{n+1} & \dots & a_{n+s-1} \\ a_{n+1} & a_n & \dots & a_{n+s-2} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n+s-1} & a_{n+s-2} & \dots & a_n \end{vmatrix}.$$ (1.12) In this study of q-geometric function theory, coefficient bounds extend beyond classical generalizations by capturing how geometric properties of analytic functions deform under the parameter q. Sharp coefficient inequalities encode precise information about the image domain and, in applied contexts, about the physical or computational systems modeled by conformal maps. For example, in medical imaging, structures with spiral or helical features can be effectively modeled using q-starlike mappings, where incorporating q-coefficient constraints improves reconstruction stability in limited-data settings; refer to [8, 22]. From a theoretical perspective, replacing integers n with q-integers $[n]_q$ and classical derivatives with Jackson's q-difference operator fundamentally alters the classes of analytic functions and their extremal behavior. Although several studies address coefficient bounds in the q-calculus, sharp estimates remain relatively scarce. Motivated by this, we establish sharp bounds for initial coefficients, Hankel and Toeplitz determinants, and functionals such as Fekete–Szegö for the newly introduced classes $\mathcal{S}_{\xi}$ and $\mathcal{S}_{\xi_a}$ . In addition, the simultaneous study of the q-class and its classical counterpart highlights their structural differences while revealing the convergence of the q-results with the classical case.

Lemma 2.1. [3] If $w(z) \in \mathcal{B}_0$ be of the form (1.3), if $b_1 > 0$ . Then $$|b_1| \le 1,$$ $|b_2| \le 1 - |b_1|^2,$ $|b_3| \le 1 - |b_1|^2 - \frac{|b_2|^2}{1 + |b_1|}.$

Lemma 2.2. [23] If $w(z) \in \mathcal{B}_0$ be of the form (1.3), if $b_1 > 0$ . Then $$b_2 = \alpha(1 - b_1^2), \quad b_3 = (1 - b_1^2) \left[ (1 - |\alpha|^2)\beta - b_1\alpha^2 \right]$$ where $\alpha, \beta \in \mathbb{C}$ with $|\alpha|, |\beta| \leq 1$ .

Lemma 2.3. [5] Let p(z) be of the form (1.2), and let $\mu \in \mathbb{C}$ . Then $$|c_2 - \mu c_1^2| \le 2 \max\{1, |2\mu - 1|\}.$$

Lemma 2.4. [20] If $w(z) \in \mathcal{B}_0$ be of the form (1.3) and $\sigma, \nu \in \mathbb{R}$ . Then the following sharp estimate exists. $$|b_3 + \sigma b_1 b_2 + \nu b_1^3| \le |\nu| \quad (\sigma, \nu) \in D_1,$$ where $$D_{1} = \begin{cases} (\sigma, \nu) : |\sigma| \ge \frac{1}{2}, & \nu \le -\frac{2}{3}(|\sigma| + 1), \\ (\sigma, \nu) : 2 \le |\sigma| \le 4, & \nu \ge \frac{1}{12}(\sigma^{2} + 8). \end{cases}$$

Lemma 2.5. [4]: If A, B, $C \in \mathbb{R}$ , let us consider $$Y(A,B,C):=\max\{|A+Bz+Cz^2|+1-|z|^2,\quad z\in\overline{\mathbb{D}}\}.$$ If $AC \geq 0$ , then $$Y(A, B, C) = \begin{cases} |A| + |B| + |C|, & |B| \ge 2(1 - |C|), \\ 1 + |A| + \frac{B^2}{4(1 - |C|)}, & |B| < 2(1 - |C|). \end{cases}$$