@phdthesis{Koch2016,
  author    = {Koch, Julia Diana},
  title     = {Value Ranges for Schlicht Functions},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-144978},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2016},
  abstract  = {This thesis deals with value sets, i.e. the question of what the set of values that a set of functions can take in a prescribed point looks like. Interest in such problems has been around for a long time; a first answer was given by the Schwarz lemma in the 19th century, and soon various refinements were proven. Since the 1930s, a powerful method for solving such problems has been developed, namely Loewner theory. We make extensive use of this tool, as well as variation methods which go back to Schiffer to examine the following questions: We describe the set of values a schlicht normalised function on the unit disc with prescribed derivative at the origin can take by applying Pontryagin's maximum principle to the radial Loewner equation. We then determine the value ranges for the set of holomorphic, normalised, and bounded functions that have only real coefficients in their power series expansion around 0, and for the smaller set of functions which are additionally typically real. Furthermore, we describe the values a univalent self-mapping of the upper half-plane with hydrodynamical normalization which is symmetric with respect to the imaginary axis can take. Lastly, we give a necessary condition for a schlicht bounded function f on the unit disc to have extremal derivative in a point z where its value f(z) is fixed by using variation methods.},
  subject      = {Pontrjagin-Maximumprinzip},
  language  = {en}
}