A Survey of Applications of the Julia Variation

This is an introductory survey on applications of the Julia variation to problems in geometric function theory. A short exposition is given which develops a method for treating extremal problems over classes F of analytic functions on the unit disk D for which appropriate subsets Fn can be constructed so that (i) F = ⋃ n Fn and (ii) for each f ∈ Fn a geometric constraint will hold that ∂f(D) will have at most n “sides”. Applications of this method which have been made to problems in the literature are reviewed, e.g., Netanyahu’s problems about the distortion theorems for starlike and convex functions constrained to contain a fixed disk; Goodman’s problems about omitted values for classes of univalent functions; integral means estimates for derivatives of convex functions; maximization problems for functionals on linear fractional transforms of convex and starlike functions.