New estimates for meromorphic functions

A boundary version of the Schwarz lemma for meromorphic functions is investigated. For the function Inf(z) = 1/z +?? k=2 knck?2zk?2, belonging to the class of W, we estimate from below the modulus of the angular derivative of the function on the boundary point of the unit disc.

Uniqueness part of Schwarz lemma for driving point impedance functions

In this paper, a boundary version of the uniqueness part of the Schwarz lemma for driving point impedance functions has been investigated. Also, more general results have been obtained for a different version of the Burns-Krantz uniqueness theorem. In these results, as different from the Burns-Krantz theorem, only the boundary points have been used as the conditions on the function. Also, more general majorants will be taken instead of power majorants in (1.1).

Applications of the Jack's lemma for the meromorphic functions at the boundary

In this paper, a boundary version of the Schwarz lemma for the class $\mathcal{% N(\alpha )}$ is investigated. For the function $f(z)=\frac{1}{z}% +a_{0}+a_{1}z+a_{2}z^{2}+...$ defined in the punctured disc $E$ such that $% f(z)\in \mathcal{N(\alpha )}$, we estimate a modulus of the angular derivative of the function $\frac{zf^{\prime }(z)}{f(z)}$ at the boundary point $c$ with $\frac{cf^{\prime }(c)}{f(c)}=\frac{1-2\beta }{\beta }$. Moreover, Schwarz lemma for class $\mathcal{N(\alpha )}$ is given.

Renormalized volumes with boundary

We develop a general regulated volume expansion for the volume of a manifold with boundary whose measure is suitably singular along a separating hypersurface. The expansion is shown to have a regulator independent anomaly term and a renormalized volume term given by the primitive of an associated anomaly operator. These results apply to a wide range of structures. We detail applications in the setting of measures derived from a conformally singular metric. In particular, we show that the anomaly generates invariant ([Formula: see text]-curvature, transgression)-type pairs for hypersurfaces with boundary. For the special case of anomalies coming from the volume enclosed by a minimal hypersurface ending on the boundary of a Poincaré–Einstein structure, this result recovers Branson’s [Formula: see text]-curvature and corresponding transgression. When the singular metric solves a boundary version of the constant scalar curvature Yamabe problem, the anomaly gives generalized Willmore energy functionals for hypersurfaces with boundary. Our approach yields computational algorithms for all the above quantities, and we give explicit results for surfaces embedded in 3-manifolds.

Uniqueness part of the Schwarz lemma at the boundary

In this paper, a boundary version of the uniqueness (or, rigidity) part of the Schwarz lemma should be investigated. Also, new results related to inner functions, inner capacities, and bilogaritmic concave majorants are obtained.

Boundary Schwarz lemma for holomorphic functions

In this paper, a boundary version of Schwarz lemma is investigated. We take into consideration a function f (z) holomorphic in the unit disc and f (0) = 0 such that ?Rf? < 1 for ?z? < 1, we estimate a modulus of angular derivative of f (z) function at the boundary point b with f (b) = 1, by taking into account their first nonzero two Maclaurin coefficients. Also, we shall give an estimate below ?f'(b)? according to the first nonzero Taylor coefficient of about two zeros, namely z=0 and z0 ? 0. Moreover, two examples for our results are considered.

Behavior of meromorphic functions at the boundary of the unit disc

In this paper, a boundary version of the Schwarz lemma for meromorphic functions is investigated. The modulus of the angular derivative of the meromorphic function Inf(z)=1/z+2nc0+3nc1z+4nc2z2+... that belongs to the class of M on the boundary point of the unit disc has been estimated from below.

A BOUNDARY VERSION OF CARTAN–HADAMARD AND APPLICATIONS TO RIGIDITY

The classical Cartan–Hadamard theorem asserts that a closed Riemannian manifold Mn with non-positive sectional curvature has universal cover [Formula: see text] diffeomorphic to ℝn, and a by-product of the proof is that [Formula: see text] is homeomorphic to Sn-1. We prove analogues of these two results in the case where Mn has a non-empty totally geodesic boundary. More precisely, if [Formula: see text], [Formula: see text] are two negatively curved Riemannian manifolds with non-empty totally geodesic boundary, of dimension n ≠ 5, we show that [Formula: see text] is homeomorphic to [Formula: see text]. We show that if [Formula: see text] and [Formula: see text] are a pair of non-positively curved Riemannian manifolds with totally geodesic boundary (possibly empty), then the universal covers [Formula: see text] and [Formula: see text] are diffeomorphic if and only if the universal covers have the same number of boundary components. We also show that the number of boundary components of the universal cover is either 0, 2 or ∞. As a sample application, we show that simple, thick, negatively curved P-manifolds of dimension ≥ 6 are topologically rigid. We include some straightforward consequences of topological rigidity (diagram rigidity, weak co-Hopf property, and the Nielson problem).