Applications of Carleman’s formula for A(z)-analytic functions

This study delves into the framework of <i>A</i>(<i>z</i>)-analytic functions, a special category of functions for which the conjugate expression <i>A</i>(<i>z</i>) is anti-analytic. Inspired by the foundational contributions of mathematicians like Grötzsch and Ahlfors – who introduced the idea of quasiconformality in function theory - our research is situated within the wider domain of complex analysis. Although quasiconformal mappings have demonstrated usefulness in fields such as gas dynamics, our investigation centers on core theoretical issues related to the characterization and recovery of analytic functions within a given domain. In particular, we focus on <i>A</i>(<i>z</i>)-analytic functions that are part of the Hardy class, known for their specific boundary characteristics and integral properties. We establish parallels to classical theorems for these functions, such as results on boundary behavior and integral formulations. A focal point of our analysis is the use and generalization of Carleman’s formula—a robust method for reconstructing analytic functions from boundary information extending the early work of G.M. Goluzin and V.I. Krylov. By means of concrete examples, we illustrate how this framework can be applied to <i>A</i>(<i>z</i>)-analytic functions, emphasizing both its theoretical significance and practical relevance in the context of complex analysis.