On the linear connectivity of the regular part of Gakhov set

A. V. Kazantsev

Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 6 (2016), pp. 55-60.

Résumé

The study of extrema of the inner mapping (conformal) radii of the plane domains has been initiated by G. Pólya and G. Szegö in connection with the isoperimetrical inequalities and by F.D. Gakhov in concern with construction of the correction classes for the exterior boundary value problems. Both traditions have been unified in the works of L. A. Aksent'ev and his successors. These works also were seminal for the systematic accumulation of the uniqueness criteria for the critical points of the conformal radii. The process of such an accumulation has led to the introduction of the Gakhov set. Let $H$ be the class of functions holomorphic in the unit disk $\mathbb D = \{\zeta\in {\mathbb C}: |\zeta| 1\}$, and let $H_0$ be the subclass of $H$ consisting of functions $f$ locally univalent in $\mathbb D$ (i.e. $f'(\zeta)\ne 0$ for all $\zeta\in \mathbb D$), normalized by the conditions $f(0) = 0$ and $f'(0) = 1$. We denote by $M_f$ the set of all critical points of the hyperbolic derivative (inner mapping radius) $h_f(\zeta) = (1 - |\zeta|^2) |f'(\zeta)|$ of the function $f\in H_0$. The geometry of the surface $h = h_f$ over the elements $a\in M_f$ is determined by the values of the index $\gamma_f(a)$ of the vector field $\nabla h_f(\zeta)$. It is well-known that if $f\in H_0$ and $a\in M_f$, then $\gamma_f(a)\in \{+1, 0, -1\}$; in particular, the equality $\gamma_f(a) = +1$ means that $\zeta = a$ is the local maximum point for the above mentioned surface. Let $k_f$ be the number of points in $M_f$. The class ${\mathcal G} = \{f\in H_0: k_f\le 1\}$ is said to be the Gakhov set, and the class ${\mathcal G}_1 = \{f\in H_0: k_f = 1, \gamma_f(M_f) = +1\}$ is called the regular part of the Gakhov set. The following question was the stimulus for our study in the present note. F. G. Avkhadiev's question. Whether the class ${\mathcal G}_1$ is a linear connected set or not? Variant of an answer is contained in the following Theorem 1. The regular part ${\mathcal G}_1$ of the Gakhov set $\mathcal G$ is a linear connected subset of the class $H$ endowed with the topology of uniform convergence on compact sets in $\mathbb D$. Proof of the above assertion is the «cascade» of reductions to the following Theorem 2. For any function $f\in {\mathcal G}_1$ there exists a family $f_t\in {\mathcal G}_1$, $t\in [0, 1]$, such that $f_0$ carries out the identity mapping, and $f_1 = f$. The family $\{f_t\}_{0\le t\le 1}$ is constructed on the base of three subfamilies. First subfamily is formed by the linear-invariant actions on $f$ by Möbius automorphisms. This subfamily connects $f$ and the function $w$ with $M_w = \{0\}$. Second subfamily is produced by the level lines of the function $w$ and connects the latter with convex function $v$. Third subfamily is convex combination of $v$ and $f_0$.

Détail

Keywords: Gakhov set, Gakhov class, linear connectivity, inner mapping radius, hyperbolic derivative, critical points.

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