A Class of Symmetric Harmonic Functions Involving a Specific q-Difference Symmetric Operator

This paper introduces a new class of harmonic functions defined through a generalized symmetric q-differential that acts on both the analytic and co-analytic parts of the function. By combining concepts from symmetric q-calculus and geometric function theory, we develop a framework that extends several well-known operators as special cases. The main contributions of this study include new criteria for harmonic univalence, sharp coefficient bounds, distortion theorems, and covering results. Our operator offers increased flexibility in modeling symmetric structures, with potential applications in complex analysis, fractional calculus, and mathematical physics. To support these theoretical developments, we provide concrete examples and highlight potential directions for future research, including extensions to higher-dimensional settings.