Gakhov set in the Merkes theorem on convex combinations
Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 156 (2014) no. 2, pp. 34-42
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The Merkes theorem deduces the starlikeness of any convex combination $f_\lambda$ of identity mapping and a holomorphic convex function $f$ in the unit disk with $f''(0)=0$. Under the same conditions, all of the functions $f_\lambda$ (except the mappings onto a strip when $\lambda=1$) are proved to belong also to the Gakhov set characterized by the property of (no more than) uniqueness of the root of the Gakhov equation. These results allow for the analogies for the exterior of the unit disk. The behavior of convex combinations is studied on the functions of the Alexander classes. For the exhaustion of each such class by the “level curves”, the “stopping moment” is found which corresponds to the exit out of the Gakhov set, and all of the trajectories of such an exit are described.
Keywords:
hyperbolic derivative, conformal radius, Gakhov set, Gakhov equation, classes of convex and starlike functions, Alexander classes.
@article{UZKU_2014_156_2_a3,
author = {T. V. Zharkova and A. V. Kazantsev},
title = {Gakhov set in the {Merkes} theorem on convex combinations},
journal = {U\v{c}\"enye zapiski Kazanskogo universiteta. Seri\^a Fiziko-matemati\v{c}eskie nauki},
pages = {34--42},
publisher = {mathdoc},
volume = {156},
number = {2},
year = {2014},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/UZKU_2014_156_2_a3/}
}
TY - JOUR AU - T. V. Zharkova AU - A. V. Kazantsev TI - Gakhov set in the Merkes theorem on convex combinations JO - Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki PY - 2014 SP - 34 EP - 42 VL - 156 IS - 2 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/UZKU_2014_156_2_a3/ LA - ru ID - UZKU_2014_156_2_a3 ER -
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T. V. Zharkova; A. V. Kazantsev. Gakhov set in the Merkes theorem on convex combinations. Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 156 (2014) no. 2, pp. 34-42. http://geodesic.mathdoc.fr/item/UZKU_2014_156_2_a3/