Geodesic
Gakhov Set in the Hornich Space under the Bloch Restriction on Pre-Schwarzians
Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 155 (2013) no. 2, pp. 65-82

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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.
Keywords: hyperbolic derivative, conformal (inner mapping) radius, bifurcations of critical points, pre-Schwarzian, Gakhov set, interior and boundary of a set.
Mots-clés : Hornich space, Bloch space 
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     author = {A. V. Kazantsev},
     title = {Gakhov {Set} in the {Hornich} {Space} under the {Bloch} {Restriction} on {Pre-Schwarzians}},
     journal = {U\v{c}\"enye zapiski Kazanskogo universiteta. Seri\^a Fiziko-matemati\v{c}eskie nauki},
     pages = {65--82},
     publisher = {mathdoc},
     volume = {155},
     number = {2},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/UZKU_2013_155_2_a5/}
}
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A. V. Kazantsev. Gakhov Set in the Hornich Space under the Bloch Restriction on Pre-Schwarzians. Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 155 (2013) no. 2, pp. 65-82. http://geodesic.mathdoc.fr/item/UZKU_2013_155_2_a5/