Some Spaces of Holomorphic Functions and Their Applications

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In this dissertation, we consider several problems in complex analysis. In the first part, we study about an explicit formula for the Hilbert transform. The celebrated integral transforms such as Fourier transform, Laplace transform, and Hilbert transform have tremendous applications in various branches of science and engineering. However, unlike to Fourier or Laplace transform, very few functions have an explicit formula for their Hilbert transforms. In this dissertation, we obtain an explicit formula for the Hilbert transform of log|f|, for the function f in Nevanlinna class having continuous extension to the real line. This family is the largest possible for which such a formula for the Hilbert transform of log|f| can be obtained. The formula is very general and implies several previously known results.
Supervisor: Srivastava, Rajesh Kumar