Coefficient body for nonlinear resolvents

Mark Elin, Fiana Jacobzon

Abstract


This paper is devoted to the study of families of so-called nonlinear resolvents. Namely, we construct polynomial transformations which map the closed unit polydisks onto the coefficient bodies for the resolvent families. As immediate applications of our results we present a covering theorem and a sharp estimate for the Schwarzian derivative at zero on the class of resolvents.

Keywords


Holomorphic function; infinitesimal generator; nonlinear resolvent; Schur parameter

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References


Berkson, E., Porta, H., Semigroups of analytic functions and composition operators, Michigan Math. J. 25 (1978), 101–115.

Comtet, L., Advanced Combinatorics, D. Reidel Publishing Co., Dordrecht, 1974.

Efraimidis, I., Vukotic, D., Applications of Livingston-type inequalities to the generalized Zalcman functional, Math. Nachr. 291 (2018), 1502–1513.

Elin, M., Shoikhet, D., Linearization Models for Complex Dynamical Systems. Topics in univalent functions, functions equations and semigroup theory, Birkhauser, Basel, 2010.

Elin, M., Shoikhet, D., Sugawa, T., Geometric properties of the nonlinear resolvent of holomorphic generators, J. Math. Anal. Appl. 483 (2020), Art. 123614.

Faa di Bruno, F., Sullo sviluppo delle Funzioni, Annali di Scienze Matematiche e Fisiche 6 (1855), 479–480.

Frabetti, A., Manchon, D., Five interpretations of Faa di Bruno’s formula, IRMA Lect. in Math. and Theor. Phys. 21 (K. Ebrahimi-Fard and F. Fauvet, eds.), Europ. Math. Soc. (2015), 91–148.

Kim, S.-A., Sugawa, T., Invariant differential operators associated with a conformal metric, Michigan Math. J. 55 (2007), 459–479.

Li, M., Sugawa, T., Schur parameters and the Caratheodory class, Results in Mathematics, accepted for publication.

Ma, W., Minda, D., Hyperbolically convex functions, Ann. Math. Polon. 60 (1994), 81–100.

Reich, S., Shoikhet, D., Metric domains, holomorphic mappings and nonlinear semigroups, Abstr. Appl. Anal. 3 (1998), 203–228.

Reich, S., Shoikhet, D., Nonlinear Semigroups, Fixed Points, and the Geometry of Domains in Banach Spaces, World Scientific Publisher, Imperial College Press, London, 2005.

Riordan, J., Combinatorial Identities, Robert E. Krieger Publishing Co., Huntington, NY, 1979.

Schur, I., Methods in Operator Theory and Signal Processing, Operator Theory: Adv. and Appl. 18, Birkhauser Verlag, 1986.

Shoikhet, D., Semigroups in Geometrical Function Theory, Kluwer, Dordrecht, 2001.

Simon, B., Orthogonal Polynomials on the Unit Circle, Part 1: Classical Theory, Colloquium Publications, Amer. Math. Society, 2005.




DOI: http://dx.doi.org/10.17951/a.2019.73.2.45-57
Date of publication: 2020-01-16 07:29:31
Date of submission: 2020-01-09 15:21:52


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