The quasihyperbolic metric, growth, and John domains.

Abstract

The research in this thesis stems from three elements of classical function theory. The first is a criterion due to Hardy and Littlewood for a function to be Holder continuous in the unit disk $\rm I\!B\subset\doubc.$ The second is a famous inequality due to Bernstein which bounds the modulus of the derivative of a polynomial in terms of the degree and the $L\sp\infty$-norm of the polynomial in $\rm I\!B$. The third is a class of domains first considered by Fritz John in his studies of plane elasticity and rigidity of local quasi-isometries. Suppose that f is a function analytic in $\rm I\!B$ and that $0<\alpha\le 1.$ Then the theorem of Hardy and Littlewood mentioned above asserts that$$\vert f\sp\prime(z)\vert\le m\ {\rm dist}(z, \partial {\rm I\!B})\sp{\alpha-1}$$for all $z\in {\rm I\!B}$ if and only if$$\vert f(z\sb1)-f(z\sb2)\vert\le{M\over\alpha}\vert z\sb1-z\sb2\vert\sp\alpha$$for all $z\sb1,\ z\sb2\in\rm I\!B,$ where m and M depend only on each other. Bernstein's inequality states that if p(z) is a polynomial of degree n, then$$\sup\sb{\rm I\!B}\vert p\sp\prime(z)\vert\le n\ \sup\sb{\rm I\!B}\vert p(z)\vert.\eqno(1)$$ A domain $D\subset\IR\sp{n}$ is a b-John domain if each pair of points $x\sb1,\ x\sb2\in D$ can be joined by an arc $\gamma\subset D$ for which$$\min\sb{j=1,2}l(\gamma(x\sb{j},y))\le b\ {\rm dist}(y, \partial D)$$for all $y\in\gamma$, where $\gamma(x\sb{j},y)$ is the subarc of $\gamma$ with endpoints $x\sb{j}$ and y. A domain is John if it is b-John for some constant b, and a simply-connected John domain in the plane is a John disk. John domains appear naturally in many areas of analysis, including complex dynamics, approximation theory, and elasticity. My research concerns the following four questions. (1) What analogues of the Hardy-Littlewood result hold when $-\infty<\alpha\le 0$? (2) What analogues of this result hold when $\rm I\!B$ is replaced by a domain $D\subset\doubc$ and $-\infty<\alpha\le 1$? (3) What analogues of this result hold when f is an arbitrary function defined in a domain $D \subset \IR\sp{n}$ and $-\infty<\alpha\le 1$? (4) For which continua $E\subset\doubc$ does an analogue of inequality (1) hold? John domains arise in examining the second, third, and fourth questions listed above.