No abstract uploaded!

No abstract uploaded!

Let$f$be a one-to-one analytic function in the unit disc with$f′$(0)=1. We prove sharp estimates for certain Taylor coefficients of the functions$(f′)$^{$p$}, where$p$<0. The proof is similar to de Branges’ proof of Bieberbach’s conjecture, and thus relies on Löwner’s equation. A special case leads to a generalization of the usual estimate for the Schwarzian derivative of$f$. We use this to improve known estimates for integral means of the functions |$f′$|^{$p$}for integers$p$⪯−2.

No abstract uploaded!

It is shown that if a function$u$satisfies a backward parabolic inequality in an open set Ω∉$R$^{$n$+1}and vanishes to infinite order at a point ($x$_{0}·$t$_{0}) in Ω, then$u(x, t$_{0})=0 for all$x$in the connected component of$x$_{0}in Ω⌢($R$^{$n$}×{$t$_{0}}).