| This thesis addresses three problems in the theory in linear functionals on families of analytic functions and the theory of quasi-conformal mappings. The basic family for the first two problems is (SIGMA), consisting of functions f that are analytic outside the unit circle, univalent, and normalized at (INFIN) so that (INFIN) is fixed and the derivative at (INFIN) is one. Such functions have expansions about (INFIN) of the form f(z) = z + b(,0) + b(,1)/z + b(,2)/z('2) + (.)(.)(.) .;The second problem looks at a subclass of (SIGMA). The subclass consists of functions which admit a quasiconformal (q.c.) extension to the interior of the unit circle with a condition on the complex dilatation. Specifically, let (SIGMA)(,c,k) = {f (ELEM) (SIGMA): f has a q.c. extension to z (LESSTHEQ) 1 that satisfies (VBAR)(VBAR)(f(,z) /f(,z)) - c(VBAR)(VBAR)(,(INFIN)) = k}, where (VBAR)c(VBAR) + k < 1. Here a technique of Lehto for the class (SIGMA)(,0,k) is extended to (SIGMA)(,c,k). An area theorem and Grunsky-type inequalities are obtained along with a result of Schiffer and Schober on b(,1) which they proved by means of a calculus of variations. Finally, a variational method is used to maxmize b(,0) for nonvanishing functions.;The last problem concerns measuring the "goodness" of q.c. mappings with prescribed boundary values. A Teichmuller mapping f is a q.c. mapping with f(,z)/f(,z) = (kappa)((phi)/(VBAR)(phi)(VBAR)) for (phi) analytic and (VBAR)(kappa) < 1. Such a mapping is extremal for its boundary values in the sense that any q.c. mapping g with the same boundary values has (VBAR)(VBAR)g(,z)/g(,z)(VBAR)(VBAR)(,(INFIN)) (GREATERTHEQ) (kappa). This measures of "goodness" is changed to a weighted sup norm and conditions on extremality are given.;The first problem is to maximize the real part of the linear functionals L(,n)(f) = (THETA)(,n) b(,1) - b(,n) on (SIGMA). It was shown by Garabedian and Schiffer that Re(2b(,1) -b(,2)) (LESSTHEQ) 2 and Re(3b(,1) -b(,3)) (LESSTHEQ) 3. It was also shown that (THETA)2 and (THETA)3 were the smallest values for (THETA)(,2) and (THETA)(,3) respectively that work. Kirwan conjectured that Re(nb(,1) -b(,n)) (LESSTHEQ) n with n being the smallest value that works. By using two results of Kubota for b(,4) and b(,5) under constraints and a subordination technique, it is shown that Re(4b(,1) -b(,4)) (LESSTHEQ) 4 and Re(3b(,1) -b(,5)) (LESSTHEQ) 3 under similar constraints. |
