Mathematical Sciences: Geometric Function Theory
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{'SignBlockName': 'John V. Ryff', 'PO_EMAI': None, 'PO_PHON': None}
Work will be done on problems related to geometric function             theory.  This expression refers to the analysis of univalent                    analytic functions of a complex variable by means of their                      geometric mapping properties.  Much of the planned work pursues                 new directions which have developed in the field since the                      resolution of the Bieberbach conjecture in 1984.  One direction                 concerns the nature of the logarithmic coefficients of a                        univalent function.  Estimates of their sums and weighted sums                  played a crucial role in the above mentioned result.  But best-                 possible inequalities were not obtained.  Work will be done in                  seeking the sharpest forms for these coefficients.  If this can                 be resolved it will provide a solution to the still unsettled                   problem of estimating consecutive coefficients of univalent                     functions.                                                                              A second thrust will be made in the direction of coefficient            regions: analyzing finite sequences of coefficients and the n-                  dimensional sets they fill out.  One specific goal is to                        demonstrate that these coefficient regions (actually, their                     convex hulls) have no ridges or spires.  This will be                           accomplished by demonstrating that a univalent function which is                extreme for two or more linear functionals must reduce to a                     rotation of a fundamental extremal function known as the Koebe                  function.                                                                               Work will also continue on questions arising in harmonic                rather than analytic mappings.  There are natural applications to               minimal surface theory in such studies.  The principal                          investigator is particularly interested in the growth and                       distortion of such maps.  Although there are a number of                        interesting questions worth considering, emphasis must be placed                on the development of an effective variational method within the                various classes of mappings.  Some progress has been made toward                this end via the Poisson representation of harmonic functions.                          Applications of this work are made to potential theory and              grid design in numerical solutions of partial differential                      equations.
{'FirstName': 'Peter', 'LastName': 'Duren', 'PI_MID_INIT': 'L', 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Peter L Duren', 'EmailAddress': 'duren@umich.edu', 'NSF_ID': '000360010', 'StartDate': '06/01/1987', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
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{'Code': '1261', 'Text': 'CLASSICAL ANALYSIS'}
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