Quotients of hyperbolic metrics (Q1689807)

The author estimates the quotient of the hyperbolic metrics of two hyperbolic domains on the Riemann sphere, one being a proper subset of the other. The bounds for the quotients are useful in complex dynamics. The purpose of this paper is to show their importance for the theory of hyperbolic metric. The quotient is shown to be close to \(1\) either on compact subsets of the proper sub-domain or near to the common boundary of the two domains. Necessary and sufficient conditions are given for the sharpness of the bounds. Altogether \(7\) examples illustrate the results.