On a Ma-Minda type class of biconvex functions via q-analytic Balloon mappings

Abstract

This paper introduces a new Ma–Minda type subclass of biconvex functions defined via subordination to a symmetric balloon-shaped domain generated by an appropriate $ \mathit{q} $-analytic mapping. The proposed subordination framework reflects the geometric influence of $ \mathit{q} $-calculus on the images of both the function and its inverse, particularly in terms of domain contraction, boundary deformation, and symmetry behavior as the parameter $ \mathit{q} $ varies. These effects provide a natural geometric deformation of the classical biconvex setting. The primary objective of this work is to investigate how the $ \mathit{q} $-parameter affects the convexity properties and analytic structure of the resulting function class. Explicit upper bounds for the initial Taylor–Maclaurin coefficients $ |\vartheta_{2}| $ and $ |\vartheta_{3}| $ are derived in closed form using $ \mathit{q} $-integers. In addition, a sharp piecewise estimate for the Fekete–Szegö functional $ |\vartheta_{3}-\varrho\, \vartheta_{2}^{2}| $ is obtained, highlighting its dependence on both the parameter $ \varrho $ and the geometry of the associated balloon-shaped domain. Furthermore, the limiting case $ \mathit{q} \to 1^{-} $ is examined, showing that the obtained results naturally reduce to the corresponding classical bounds for biconvex functions. To illustrate the geometric characteristics and applicability of the introduced class, several representative examples together with graphical visualizations of the corresponding image domains are also presented.