Geodesic
Estimates of the hyperbolic radius gradient and Schwarz–Pick inequalities for the eccentric annulus
Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 158 (2016) no. 2, pp. 172-179 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $\Omega$ and $\Pi$ be hyperbolic domains in the complex plane $\mathbb C$. By $A(\Omega,\Pi)$ we shall designate the class of functions $f$ which are holomorphic or meromorphic in $\Omega$ and such that $f(\Omega)\subset\Pi$. Estimates of the higher derivatives $|f^{(n)}(z)|$ of the analytic functions from the class $A(\Omega,\Pi)$ with the punishing factor $C_n(\Omega,\Pi)$ is one of the main problems of geometric theory of functions. These estimates are commonly referred to as Schwarz–Pick inequalities. Many results concerning this problem have been obtained for simply connected domains. Therefore, the research interest in such problems for finitely connected domains is natural. As known, the constant $C_2(\Omega,\Pi)$ for any pairs of hyperbolic domains depends only on the hyperbolic radius gradient of the corresponding domains. The main result of this paper is estimates of the hyperbolic radius gradient and the punishing factor in the Schwarz–Pick inequality for the eccentric annulus. We also consider the extreme case – the randomly punctured circle.
Keywords: Poincare metrics, Schwarz–Pick inequalities, conformal mapping, punishing factors. 
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D. Kh. Giniyatova. Estimates of the hyperbolic radius gradient and Schwarz–Pick inequalities for the eccentric annulus. Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 158 (2016) no. 2, pp. 172-179. http://geodesic.mathdoc.fr/item/UZKU_2016_158_2_a1/

[1] Goluzin G. M., Geometric Theory of Functions of a Complex Variable, Nauka, Moscow, 1966, 628 pp. (In Russian) | MR

[2] Avkhadiev F. G., Wirths K.-J., Schwarz–Pick Type Inequalities, Birkhäuser, Boston–Berlin–Bern, 2009, 156 pp. | MR | Zbl

[3] Ruscheweyh St., “Über einige Klassen in Einheitskreis holomorpher Funktionen”, Ber. Math.-Stat. Sekt. Forschungszent. Graz., 1974, no. 7, 1–12

[4] Ruscheweyh St., “Two remarks on bounded analytic functions”, Serdica (Bulg. Math. Publ.), 11:2 (1985), 200–202 | MR | Zbl

[5] Yamashita S., “Higher derivatives of holomorphic function with positive real part”, Hokkaido Math. J., 29:1 (2000), 23–36 | DOI | MR | Zbl

[6] Avkhadiev F. G., Wirths K.-J., “Schwarz–Pick inequalities for derivatives of arbitrary order”, Constr. approx., 19:1 (2003), 265–277 | DOI | MR | Zbl

[7] Avkhadiev F. G., Wirths K.-J., “The punishing factors for convex pairs are $2^{n-1}$”, Rev. Math. Iberoam., 23:3 (2007), 847–860 | DOI | MR | Zbl

[8] Avkhadiev F. G., Wirths K.-J., “Estimates of the derivatives of meromorphic maps from convex domains into concave domains”, Comp. Methods Funct. Theory, 8:1 (2008), 107–119 | DOI | MR | Zbl

[9] Li J.-L., “Estimates for derivatives of holomorphic functions in a hyperbolic domain”, J. Math. Anal. Appl., 329:1 (2007), 581–591 | DOI | MR | Zbl

[10] Chua K. S., “Derivatives of univalent functions and the hyperbolic metric”, Rocky Mt. J. Math., 26:1 (1996), 63–75 | DOI | MR | Zbl

[11] Giniyatova D. Kh., “Generalization of theorems of Szász and Ruscheweyh on exact bounds for derivatives of analytic functions”, Russ. Math., 53:12 (2009), 72–76 | DOI | MR | Zbl

[12] Abramov D. A., Avkhadiev F. G., Giniyatova D. Kh., “Versions of the Schwarz lemma for domain moments and the torsional rigidity”, Lobachevskii J. Math., 32:2 (2011), 149–158 | DOI | MR | Zbl

[13] Avkhadiev F. G., Wirths K.-J., “Punishing factors for finitely connected domains”, Monatsh. Math., 147:2 (2006), 103–115 | DOI | MR | Zbl

[14] Avkhadiev F. G., Wirths K.-J., A theorem of Teichmüller, Uniformly Perfect Sets and Punishing Factors, Preprint, Tech. Univ. Braunschweig, Braunschweig, 2005, 16 pp.

[15] Kazantsev A. V., “Gakhov set in the Hornich space under the Bloch restriction on pre-Schwarzians”, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 155, no. 2, 2013, 65–82 (In Russian) | MR | Zbl

[16] Kazantsev A. V., “On the exit out of the Gakhov set controlled by the subordination conditions”, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 156, no. 1, 2014, 31–43 (In Russian) | MR

[17] Avkhadiev F. G., Wirths K.-J., “The conformal radius as a function and its gradient image”, Isr. J. Math., 145:1 (2005), 349–374 | DOI | MR | Zbl

[18] Aksentyev L. A., Akhmetova A. N., “On the gradient of the conformal radius of a plane domain”, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 154, no. 1, 2012, 167–178 (In Russian) | MR

[19] Akhmetova A. N., Properties of the conformal radius and the uniqueness theorem for external inverse boundary value problems, Cand. Phys.-Math. Sci. Diss., Kazan, 2009