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Title:  Extremal functions related to convexity and martingales 
Author(s):  Suwannaphichat, Sineenuch 
Director of Research:  Hinkkanen, Aimo 
Doctoral Committee Chair(s):  Miles, Joseph B.; Loeb, Peter A. 
Doctoral Committee Member(s):  Hinkkanen, Aimo; Merenkov, Sergiy A. 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Extremal functions
Convex functions 
Abstract:  We consider the problem of finding the extremal function in the class of realvalued biconvex functions satisfying a boundary condition on a product of the unit ball with itself, with a suitable norm in the plane. We want to maximize the biconvex function at a point in the domain where the second component is fixed and therefore we can consider the biconvex function as a convex function of the first component. We then find a representation for the convex function in terms of some functions of a suitable quotient of second order partial derivatives of the convex function, where these functions will satisfy certain conditions so that the biconvex function will have the given boundary values. From the quotient of second order partial derivatives of the convex function, we obtain a relation leading to the Hopf differential equation, whose solution involves a parameter function. With a given boundary function, we perform a variation of the parameter function by a small realvalued function. Then we find the change of the representation of the convex function. If the convex function is an extremal function, then the rate of change with respect to the variation made to the parameter function is zero. This will be the condition that we are looking for in an extremal function. 
Issue Date:  20140916 
URI:  http://hdl.handle.net/2142/50350 
Rights Information:  Copyright 2014 Sineenuch Suwannaphichat 
Date Available in IDEALS:  20140916 20160922 
Date Deposited:  201408 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois