Harmonic Mappings and Harmonic Hardy Spaces

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 Sheil-Small 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 complex-valued 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 Riesz-Fej\(\acute{e}\)r inequality for harmonic functions, for the special case \(p=2\).