Gabriel-Type Estimates for Harmonic Quasiregular Mappings and Stoilow Classes

Abstract

We establish a dilatation-dependent Gabriel inequality for sense-preserving harmonic $K$-quasiregular mappings in $h^2$. If $f=h+\overline g\in h^2$ satisfies $|g'|\le k|h'|$, where $K=(1+k)/(1-k)$, then every convex curve $\Gamma\subset\D$ satisfies \[ \int_\Gamma |f(z)|^2\ds(z) \le 2\frac{(1+k)^2}{1+k^2} \int_\T |f^*(\zeta)|^2\,|d\zeta| =\frac{4K^2}{K^2+1} \int_\T |f^*(\zeta)|^2\,|d\zeta|. \] The coefficient recovers the sharp analytic constant $2$ at $K=1$, is strictly smaller than the general harmonic constant $4$ for every finite $K$, and tends to $4$ as $K\to\infty$. The proof retains quantitative information on the analytic and co-analytic parts through the Hardy--Stein identity and the complex dilatation bound. To place this estimate in a broader quasiregular setting, we develop a maximal-function principle based on the Carleson-measure geometry of convex curves. We also identify a log-subharmonic modulus class that inherits the sharp analytic constant and prove Stoilow-factorization criteria under explicit boundary-weight and cone-distortion assumptions. These results distinguish distortion-dependent estimates from those arising through general boundary maximal control. The resulting formulation clarifies which conclusions follow from harmonic quasiregularity and which require separate boundary regularity assumptions.