Gakhov Set in the Hornich Space under the Bloch Restriction on Pre-Schwarzians

Gakhov set contains exactly those functions in the Hornich space over the unit disk which have the unique critical point of the conformal radius. The position of the intersection $\mathcal{A}$ of the Gakhov set and the Bloch space $\mathcal{B}$ is studied relative to the Banach structure of $\mathcal{B}$. A connection is revealed between the topological characteristics of the set $\mathcal{A}$ and the values of the curvature and index of the critical points for the functions in $\mathcal{A}$. An effective description is given for the set of points on the boundary of $\mathcal{A}$ with minimal pre-norm. By using the Minkowski functional, the starlikeness of the subset of the functions in $\mathcal{A}$ with the zero critical point of the conformal radius is established.