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.

Let$R$be a Noetherian ring, $\mathfrak{a}$ an ideal of$R$,$M$an$R$-module and$n$a non-negative integer. In this paper we first study the finiteness properties of the kernel and the cokernel of the natural map $f\colon\operatorname{Ext}^{n}_{R}(R/\mathfrak{a},M)\to \operatorname{Hom}_{R}(R/\mathfrak{a},\mathrm{H}^{n}_{\mathfrak{a}}(M))$ , under some conditions on the previous local cohomology modules. Then we get some corollaries about the associated primes and Artinianness of local cohomology modules. Finally we will study the asymptotic behavior of the kernel and the cokernel of the natural map in the graded case.

In this article, we study the Hamiltonian non-displaceability of Gauss images of isopara- metric hypersurfaces in the spheres as Lagrangian submanifolds embedded in complex hyperquadrics.