A SURVEY OF APPLICATIONS OF THE JULIA VARIATION ROGER W. BARNARD, KENT PEARCE * AND CAROLYNN CAMPBELL † Abstract. 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. Key words. Julia variation, variational methods, geometric function theory AMS subject classifications. 30C This is a survey paper on the applications of a variational method introduced by J. Krzyz in [29]. It is based on Julia’s modification of Hadamard’s variation of the Green’s function. In the early 1900’s, in order to investigate the behavior of Green’s function for slight deformations on the boundary of its domain, Hadamard developed his variational formulas [23]. The validity of the formulas relied on the domain having (at least) a continuously differentiable or smooth boundary. In the 1930’s Julia, modifying Hadamard’s procedure, derived a new variational formula in terms of the Riemann mapping function for the domain. This also required the boundary of the domain to be smooth. This requirement proved to be too restrictive to solve most extremal problems since the corresponding extremal domains have non- smooth, typically piece-wise smooth, boundaries. In particular, Schiffer suggested in [39] that the requirement of a smooth boundary was too restrictive for extremal problems for the general class of univalent functions; he then proceeded to develop his method of interior variations to circumvent the issue. The first author was introduced to the Hadamard and Julia variational formulas during his graduate studies at the University of Maryland by J. Hummel. Hummel had used results on the Julia variation, obtained in [38], to create in [27] a general variational method for starlike functions. He gave a discussion of the Hadamard and Julia formulas in his lecture notes in [26]. We note also that Julia’s formula was used by M.S. Robertson in [37] to develop a variational method for analytic functions on the unit disk with positive real part. Aside from Schiffer’s and Hummel’s references, the Hadamard and Julia vari- ational formulas remained generally dormant until Krzyz in [29] applied the Julia variational formula to convex polygons. A rigorous proof that the formula is valid when the boundary contains corners was provided by Barnard and Lewis in [10]. It was found later that an independent proof had been given earlier by Warshawski in [44]. Consequently, the initial condition of smoothness on the boundary of the domain could be replaced by a piece-wise smooth condition. This allowed for the method to be applied to a fairly large class of functions; in particular, to any class of functions whose image domains could be approximated by domains bounded by a finite number * Department of Mathematics and Statistics, Texas Tech University, Lubbock, Texas 79409 (barnard@math.ttu.edu, pearce@math.ttu.edu) † Department of Mathematics, Austin Community College, Austin, Texas 1