Harmonic Mappings and Harmonic Hardy Spaces
Speaker 
Dr. Sairam Kaliraj
ISI, Chennai


When 
Mar 01, 2018
from 04:00 PM to 05:00 PM 
Where  LH 006 
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Abstract: The Bieberbach conjecture has been the driving force behind the development of geometric function theory. The harmonic analogue of Bieberbach conjecture due to Clunie and SheilSmall deals with the upper bounds of the Taylor coefficients of univalent harmonic functions \(f(z)\)=\(h(z)\)+\(\overline{g(z)}\)= (z+\sum^\infty_{n=2}\) \(a_nz^n +\overline{\sum^\infty_{n=2} b_nz^n}\) defined on the open unit disk \(z<1\). This coefficient conjecture is still open even for \(a_2\). In this talk, we present our results on this conjecture and its various applications, which includes the growth of \(f(z)\), the growth of the determinant of the Jacobian \(J_f(z)\), and the approximation of univalent harmonic mapping \(f(z)\) by univalent harmonic polynomials. Then, we discuss about the Riesz  Fej\(\acute{e}\)r inequality for complexvalued harmonic functions in the harmonic Hardy space \({\bf h}^p\) for all \(p > 1\). The result is sharp for \(p \in (1,2]\). Moreover, we prove two variant forms of RieszFej\(\acute{e}\)r inequality for harmonic functions, for the special case \(p=2\).