Quantum-deformed growth geometry in the open unit disk, with applications in subordination, extremal functions, and phase transitions

Purpose This paper introduces a new (q, τ)–balanced growth class of analytic functions in the unit disk, motivated by quantum–deformation computing and scale–dependent geometric analysis. The aim is to develop a geometric framework that simultaneously captures radial growth and angular distortion through finite–scale (q, τ)–deformations. Design/methodology/approach The proposed class is defined via a nonlinear admissibility condition involving a (q, τ)–deformed logarithmic growth factor. Unlike classical starlike and spiral-like families characterized by half–plane constraints, the new class is governed by a parabolic admissible region incorporating quadratic damping effects. A generalized Jack lemma in the (q, τ)–setting is established and applied to derive a sharp subordination theorem with an explicit dominant mapping. The theoretical analysis is complemented by extremal function techniques and conformal visualization of the dominant mappings. Findings The developed approach yields explicit starlikeness radius results together with sharp extremal functions that attain the admissibility boundary. A geometric phase diagram in the parameter space (α, β, q, τ) is obtained, separating regions of full radial coherence, critical transition, and loss of starlikeness. The conformal plots demonstrate how the deformation parameters contract the admissible domain and influence extremal directions. Several known classical results are recovered as limiting cases of the proposed framework. Originality/value The paper introduces a novel (q, τ)–balanced growth structure combining quantum–deformation concepts, nonlinear admissibility methods, and geometric function theory within a unified setting. The use of a parabolic admissibility region together with finite–scale logarithmic deformation provides a new perspective on analytic growth phenomena and extends classical starlike theory to a broader nonlocal geometric regime.