the schwarz lemma
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2021 ◽  
Vol 109 (123) ◽  
pp. 153-162
Author(s):  
Bülent Örnek

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.


Author(s):  
Ole Fredrik Brevig ◽  
Joaquim Ortega-Cerdà ◽  
Kristian Seip

Filomat ◽  
2021 ◽  
Vol 35 (5) ◽  
pp. 1511-1517
Author(s):  
Gradimir Milovanovic ◽  
Abdullah Mir ◽  
Abrar Ahmad

In this paper, we obtain certain sharp estimates for the maximal modulus of a rational function with prescribed poles. The proofs of the obtained results are based on the new version of the Schwarz lemma for regular functions which was suggested by Osserman. The obtained results produce several inequalities for polynomials as well.


Author(s):  
SHANE ARORA ◽  
HAZEL BROWNE ◽  
DANIEL DANERS

We discuss an alternative approach to Fréchet derivatives on Banach spaces inspired by a characterisation of derivatives due to Carathéodory. The approach allows many questions of differentiability to be reduced to questions of continuity. We demonstrate how that simplifies the theory of differentiation, including the rules of differentiation and the Schwarz lemma on the symmetry of second-order derivatives. We also provide a short proof of the differentiable dependence of fixed points in the Banach fixed point theorem.


2020 ◽  
Vol 14 (1) ◽  
pp. 150-168
Author(s):  
Miodrag Mateljevic ◽  
Marek Svetlik

We give simple proofs of various versions of the Schwarz lemma for real valued harmonic functions and for holomorphic (more generally harmonic quasiregular, shortly HQR) mappings with the strip codomain. Along the way, we get a simple proof of a new version of the Schwarz lemma for real valued harmonic functions (without the assumption that 0 is mapped to 0 by the corresponding map). Using the Schwarz-Pick lemma related to distortion for harmonic functions and the elementary properties of the hyperbolic geometry of the strip we get optimal estimates for modulus of HQR mappings.


Filomat ◽  
2020 ◽  
Vol 34 (11) ◽  
pp. 3711-3720
Author(s):  
Marek Svetlik

In this note we consider some generalizations of the Schwarz lemma for harmonic functions on the unit disk, whereby values of such functions and the norms of their differentials at the point z = 0 are given.


Filomat ◽  
2020 ◽  
Vol 34 (9) ◽  
pp. 2953-2959
Author(s):  
Nafi Örnek ◽  
Timur Düzenli

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).


2019 ◽  
Vol 38 (7) ◽  
pp. 219-226
Author(s):  
Tugba Akyel ◽  
Bulent Nafi Ornek

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.


2019 ◽  
Vol 100 (1) ◽  
pp. 86-96 ◽  
Author(s):  
NAK EUN CHO ◽  
BOGUMIŁA KOWALCZYK ◽  
ADAM LECKO

We apply the Schwarz lemma to find general formulas for the third coefficient of Carathéodory functions dependent on a parameter in the closed unit polydisk. Next we find sharp estimates of the Hankel determinant $H_{2,2}$ and Zalcman functional $J_{2,3}$ over the class ${\mathcal{C}}{\mathcal{V}}$ of analytic functions $f$ normalised such that $\text{Re}\{(1-z^{2})f^{\prime }(z)\}>0$ for $z\in \mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$, that is, the subclass of the class of functions convex in the direction of the imaginary axis.


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