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Harmonic Mappings and Harmonic Hardy Spaces

Dr. Sairam Kaliraj ISI, Chennai
 Speaker Dr. Sairam Kaliraj ISI, Chennai Mar 01, 2018 from 04:00 PM to 05:00 PM LH 006 vCal iCal

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 $$p=2$$.

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