Representations for the Bloch Type Semi-norm of Fréchet Differentiable Mappings

Abstract

In this paper we give some results concerning Fréchet differentiable mappings between domains in normed spaces with controlled growth. The results are mainly motivated by Pavlović's equality for the Bloch semi-norm of continuously differentiable mappings in the Bloch class on the unit ball of the Euclidean space as well as the very recent Jocić's generalization of this result.

On holomorphic functions on negatively curved manifolds

“…(see [10] for a similar result in a general setting, i.e., for Fréchet differentiable mappings between vector spaces with norm). In the sequel we will consider a generalization of the Pavlović result for holomorphic functions on the manifold M .…”

“…|ζ−η| = ω(ζ) (for this equality in a general setting, i.e., for vector spaces with norm, we refer to [10]), from the last inequality we obtain (2.2).…”

Applications of the algebraic geometry of the Putman–Wieland conjecture

“…Here is the slight improvement on [14, Theorem 1.5], which we will prove in Subsection 5.10. We let $$\lambda _1(X)$$ denote the smallest nonzero eigenvalue of the Laplacian acting on $$L^2$$ functions on $$X$$.…”

“…Remark Theorem 1.6 is slightly stronger than [14, Theorem 1.5] as we now explain. In their result, they first choose $$\varepsilon$$ and then choose $$g$$ subject to the inequality $$g \geqslant (1+\epsilon )/\epsilon$$, or equivalently $$\varepsilon \geqslant \frac{1}{g-1}$$.…”

“…In [14, Theorem 1.5], Marković–Tošić verified some new cases of the Putman–Wieland conjecture. We are able to recover their results from ours.…”