Coefficient Bounds and Parameter Geometry for Gamma-Deformed Mathieu–Ma–Minda Bi-Univalent Functions

The factorial Mathieu multiplier used in the nearest bi-univalent model is substituted with a gamma-shifted family using deformation parameter $\tau \ge 0$. In this case, one differential operator describes all class operators introduced previously, whereas the function and inverse subordination can be controlled by two different generalized Ma–Minda functions. Explicit bounds for $|a_2|$, $|a_3|$ and the Fekete–Szegő functional follow from identities involving exact second-order coefficients. These are sharpened by using the full Schwarz–Pick estimate $\left|{\omega }_2\right| \le |1-|{\omega }_1{|}^2$. The estimates continue to hold even when $\mathcal{Q}=0$. The positive-real-part, strongly starlike, Janowski, mixed Mathieu, and phase-dependent Noshiro families are included with explicit admissibility criteria, except for the Noshiro family, which has its phase limited to $-\pi <\varphi <\pi$, since ${\Delta }_2$ vanishes at the excluded endpoint. The numerical analysis compares the gamma deformation with the factorial case, partitions parameter space according to the active coefficient estimate, locates $\mathcal{Q}=0$, and shows how unequal targets displace the center of the Fekete–Szegő bound. The auxiliary Schwarz inequalities are sharp, but simultaneous equality within the full bi-univalent class is not established.