On the preservation of generalized reduced modulus under some geometric transformations of domains in the plane

We study necessary and sufficient conditions for the preservation of the generalized reduced modulus under the extension of domain or a part of its boundary. Also, we give such conditions for polarization and dissymmetrization. As applications, the descriptions of extremal configurations in the known inequalities for the inner radii, the Green functions and the Robin functions are obtained. Some of our results supplement the known boundary distortion theorems for univalent regular functions.