Two Coefficient Conjectures for Nonvanishing Hardy Functions, I

There are two eminent still open conjectures for nonvanishing holomorphic Hardy functions f(z) on the unit disk. The Hummel–Scheinberg–Zalcman conjecture posed in 1977 extends Krzyz’s of 1968 to Hp spaces with finite p > 1 and states that Taylor’s coefficients f ? norm ?f?p ? sharply estimated by |cn| (2/e)1 ? 1/p, appropriate extremal functions. Both have been investigated many authors; however remain open. desired estimate 2/e H? ?f?? was established only initial cn n 5 known results it is true = 2 as well some special subclasses Hp. We prove here all H2m m N. In limit ? ?, this also provides proof conjecture. Our approach involves deep from Teichmüller space theory, especially Bers isomorphism theorem punctured Riemann surfaces, quasiconformal deformations