# On holomorphic functions on negatively curved manifolds

@article{Markovic2021OnHF, title={On holomorphic functions on negatively curved manifolds}, author={Marijan Markovic}, journal={Monatshefte f{\"u}r Mathematik}, year={2021} }

Based on a well known Sh.-T. Yau theorem we obtain that the real part of a holomorphic function on a Kähler manifold with the Ricci curvature bounded from below by −1 is contractive with respect to the distance on the manifold and the hyperbolic distance on (−1, 1) inhered from the domain (−1, 1)×R. Moreover, in the case of bounded holomorphic functions we prove that the modulus is contractive with respect to the distance on the manifold and the hyperbolic distance on the unit disk.

#### References

SHOWING 1-10 OF 26 REFERENCES

On harmonic functions and the hyperbolic metric

- Mathematics
- 2013

Motivated by some recent results of Kalaj and Vuorinen (Proc. Amer. Math. Soc., 2012), we prove that positive harmonic functions defined in the upper half--plane are contractions w.r.t. hyperbolic… Expand

Hyperbolic metric on the strip and the Schwarz lemma for HQR mappings

- Mathematics
- 2018

We give simple proofs of various versions of the Schwarz lemma for real valued harmonic functions and for holomorphic (more generally harmonic quasi\-re\-gu\-lar, shortly HQR) mappings with the strip… Expand

Function Classes on the Unit Disc: An Introduction

- Mathematics
- 2019

The monograph contains a study on various function classes, a number of new results and new or easy proofs of old result (Fefferman-Stein theorem on subharmonic behavior, theorem on conjugate… Expand

Schwarz lemma and Kobayashi metrics for harmonic and holomorphic functions

- Mathematics
- Journal of Mathematical Analysis and Applications
- 2018

Abstract In this note we mainly consider various version of Schwarz lemma and its relatives related to harmonic and holomorphic functions including several variables. It turns out that our methods… Expand

Invariant gradient in refinements of Schwarz and Harnack inequalities

- Mathematics
- 2018

In this paper we prove a refinement of Schwarz’s lemma for holomorphic mappings from the unit ball Bn ⊂ Cn to the unit disk D ⊂ C obtained by Kalaj in [3]. We also give some corollaries of this… Expand

An extension of Schwarz’s lemma

- Mathematics
- 1938

and has the constant curvature -4. 2. Consider now an analytic function o =f(z) from the circle I zx < 1 to a Riemann surface W. The analyticity is expressed by the fact that every local parameter w… Expand

A Schwarz lemma for the modulus of a vector-valued analytic function

- Mathematics
- 2011

It is proved that |∇|f|(z)|≤1-|f(z)| 2 /1-|z| 2 , z∈D, where f: D ↦ B k is an analytic function from the unit disk D into the unit ball B k C ℂ k . Applications to the Lipschitz condition of the… Expand

SCHWARZ LEMMA FOR HOLOMORPHIC MAPPINGS IN THE UNIT BALL

- Mathematics
- Glasgow Mathematical Journal
- 2017

Abstract In this note, we establish a Schwarz–Pick type inequality for holomorphic mappings between unit balls B n and B m in corresponding complex spaces. We also prove a Schwarz-Pick type… Expand

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

- Mathematics
- 2020

In this paper we give some results concerning Frechet differentiable mappings between domains in normed spaces with controlled growth. The results are mainly motivated by Pavlovic’s equality for the… Expand

Lipschitz constants for the real part and modulus of analytic mappings on a negatively curved surface

- Mathematics
- 2020

We prove that if f , $$|\mathfrak {R}f|<1$$ | R f | < 1 , is an analytic mapping on a surface $$\Sigma $$ Σ with curvature bounded from below by a constant $$k<0$$ k < 0 , and if $$\sigma $$ σ is the… Expand