A generalized Schwarz–Pick inequality for quasiconformal mappings satisfying Poisson's equation

Zhong, Deguang

doi:10.1216/rmj.2021.51.1133

Cited by 2 publications

(4 citation statements)

References 22 publications

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“…According to Lemma 2.2, we can get a Schwarz-Pick type inequality for mappings satisfying the Poisson's equation. This kind of lemma has been established in [25]. For the completeness, we will give the proof here.…”

mentioning

confidence: 85%

“…The hyperbolically partial derivative on

f_{z}

of a quasiconformal mapping f from domain D of hyperbolic type into domain

D^{'}

of hyperbolic type is defined as

false| false| \partial f false| false| = \frac{λ_{D^{'}} \circ f}{λ_{D}} \cdot false| f_{z} false| .

In [19], Knežević and Mateljević obtained the following estimate on hyperbolically partial derivative of K ‐quasiconformal harmonic self‐mappings of

double-struckD

. This result has been generalized in [4, 6, 25, 28]. Theorem A Let f be a K ‐quasiconformal harmonic self‐mapping of

double-struckD

.…”

Section: Some Preparationsmentioning

confidence: 99%

“…In [19], Knežević and Mateljević obtained the following estimate on hyperbolically partial derivative of K-quasiconformal harmonic self-mappings of D. This result has been generalized in [4,6,25,28].…”

mentioning

confidence: 96%

See 2 more Smart Citations

On a Quasi‐isometry for Quasiconformal Mapping Satisfying Poisson Differential Inequality

Zhong

2020

Mathematika

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mentioning

confidence: 85%

“…The hyperbolically partial derivative on

f_{z}

of a quasiconformal mapping f from domain D of hyperbolic type into domain

D^{'}

of hyperbolic type is defined as

false| false| \partial f false| false| = \frac{λ_{D^{'}} \circ f}{λ_{D}} \cdot false| f_{z} false| .

In [19], Knežević and Mateljević obtained the following estimate on hyperbolically partial derivative of K ‐quasiconformal harmonic self‐mappings of

double-struckD

. This result has been generalized in [4, 6, 25, 28]. Theorem A Let f be a K ‐quasiconformal harmonic self‐mapping of

double-struckD

.…”

Section: Some Preparationsmentioning

confidence: 99%

See 1 more Smart Citation

On a Quasi‐isometry for Quasiconformal Mapping Satisfying Poisson Differential Inequality

Zhong

2020

Mathematika

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“…It is worth pointing out that Dieudonné's Lemma and Rogosinski's Lemma are the limiting cases of Mercer's result as z → z 0 and z 0 → 0 respectively. In recent years, more and more articles on regions of variability and Schwarz-Pick type estimates have been written [18,19,27,28,29].…”

mentioning

confidence: 99%

Estimates of the Second Derivative of Bounded analytic Functions

Chen

2019

Bull. Aust. Math. Soc.

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Let z 0 and w 0 be given points in the open unit disk D with |w 0 | < |z 0 |, and H 0 be the class of all analytic self-maps f of D normalized by f (0) = 0. In this paper, we establish the third order Dieudonné Lemma, and apply it to explicitly determine the variability region {f ′′′ (z 0 ) : f ∈ H 0 , f (z 0 ) = w 0 , f ′ (z 0 ) = w 1 } for given z 0 , w 0 , w 1 and give the form of all the extremal functions.2010 Mathematics Subject Classification. Primary 30C80; secondary 30F45.

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A generalized Schwarz–Pick inequality for quasiconformal mappings satisfying Poisson's equation

Cited by 2 publications

References 22 publications

On a Quasi‐isometry for Quasiconformal Mapping Satisfying Poisson Differential Inequality

On a Quasi‐isometry for Quasiconformal Mapping Satisfying Poisson Differential Inequality

Estimates of the Second Derivative of Bounded analytic Functions

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