Sharp Bounds for $q$-Starlike Functions and Their Classical Counterparts

TL;DR

The paper addresses sharp coefficient and determinant problems for Ma-Minda type $q$-starlike functions, introducing the classes $\mathcal{S}^*_{\xi_q}$ and its classical limit $\mathcal{S}^*_{\xi}$ via subordination to $\xi_q$ and $\xi$ and using the Jackson $q$-derivative $d_q$. Leveraging $q$-calculus, subordination, and convolution representations, it derives explicit sharp bounds for initial coefficients ($|a_2|$, $|a_3|$, $|a_4|$), Fekete–Szegö, Kruskal, Zalcman-type functionals, and Hankel/Toeplitz determinants, with extremal functions $\tilde f_q$ and $\tilde f$ certifying sharpness. The results continuously connect the $q$-analogue to the classical theory as $q\to1$, and the extremals are characterized by convolution equations, providing a constructive view of the deformation from quantum/discrete to classical settings. This framework offers a unified path to extend $q$-analogs to other Ma-Minda classes and to higher-order coefficient problems in geometric function theory.

Abstract

Geometric function theory increasingly draws on $q$-calculus to model discrete and quantum-inspired phenomena. Motivated by this, the present paper introduces two new subclasses of analytic functions: the class $\mathcal{S}^{*}_{ξ_q}$ of $q$-starlike functions associated with the Ma-Minda function $ξ_q(z)$, and its classical counterpart $\mathcal{S}^{*}_ξ$ associated with $ξ(z)$, where $q \in (0,1)$. We conduct a systematic investigation of the geometric properties of these function classes and establish sharp coefficient estimates, including Fekete-Szegö, Kruskal, and Zalcman-type inequalities. Furthermore, we obtain sharp bounds of Hankel and Toeplitz determinants for both classes.

Figures (4)

• Figure 1: Image domain $\xi_{0.8}(\mathbb{D})$.
• Figure 2: Image domain $\xi(\mathbb{D})$.
• Figure 3: Plot of $\varphi_1(b_1, q)$ for $b_1, q \in (0,1)$.
• Figure 4: Plot of $\varphi_2(b_1, q)$ for $b_1, q \in (0,1)$.

Theorems & Definitions (36)

• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Corollary 1