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In this note we establish an advanced version of the inverse function theorem and study some local geometrical properties like starlikeness and hyperbolic convexity of the inverse function under natural restrictions on the numerical range of the underlying mapping.
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.
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