Geodesic
Generalization of theorems of Sz\'asz and Ruscheweyh on exact bounds for derivatives of analytic functions
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 12 (2009), pp. 84-89

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Let $\Omega$ and $\Pi$ be two domains in the extended complex plane equipped by the Poincaré metric. In this paper we obtain analogs of Schwarz–Pick type inequalities in the class $A(\Omega,\Pi)=\{f\colon\Omega\to\Pi\}$ of functions locally holomorphic in $\Omega$; for the domain $\Omega$ we consider the exterior of the unit disk and the upper half-plane. The obtained results generalize the well-known theorems of Szasz and Ruscheweyh about the exact estimates of derivatives of analytic functions defined on the disk $|z|1$.
Keywords: Schwarz–Pick type inequalities, analytic functions, Poincaré metric. 
@article{IVM_2009_12_a10,
     author = {D. Kh. Giniyatova},
     title = {Generalization of theorems of {Sz\'asz} and {Ruscheweyh} on exact bounds for derivatives of analytic functions},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {84--89},
     publisher = {mathdoc},
     number = {12},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2009_12_a10/}
}
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D. Kh. Giniyatova. Generalization of theorems of Sz\'asz and Ruscheweyh on exact bounds for derivatives of analytic functions. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 12 (2009), pp. 84-89. http://geodesic.mathdoc.fr/item/IVM_2009_12_a10/