Mathematical Sciences: Value Distribution of Complex Functions

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Two of the main objectives of this work in the mathematical theory of entire and meromorphic functions are to give sharp bounds for the sum of Nevanlinna deficiencies of a meromorphic function and to analyze the minimum modulus of an entire function. The deficiency of a meromorphic function is a measure of the function's affinity for a given complex number. Generally the deficiency is infinite, that is, meromorphic functions achieve most complex values infinitely often. For functions of finite order the sum of the defects is known to be no greater than two. The amount this falls short of two should be measured in terms of the order of the function. The conjectured upper bounds (proposed many years ago by Albert Edrei) have been proved to be valid for large subclasses of functions. There is considerable evidence to suggest that a complete solution of this fundamental estimate may be near at hand. The second thrust of this project seeks to compare the maximum and minimum moduli of entire (or subharmonic) functions of prescribed order. The moduli are compared for large values of the underlying variable. When the order of the function is less than one, the classic result of Wiman-Valiron gives a sharp lower bound. For order equal to unity or above little exact information is known. New, related, results may provide information related to this question. The focus of the research will be on relating the number and distribution of zeros to the extremal cases. This work is expected to have ancillary applications to complex differential equations and quasiconformal mapping.