Schwarz-Type Lemma, Landau-Type Theorem, and Lipschitz-Type Space of Solutions to Inhomogeneous Biharmonic Equations

Chen, Shaolin; Li, Peijin; Wang, Xiantao

doi:10.1007/s12220-018-0083-6

Cited by 18 publications

(9 citation statements)

References 40 publications

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“…Biharmonic equations arise in many physical situations, particularly in fluid dynamics and elasticity problems (cf previous studies()). Chen et al studied the properties of inhomogeneous biharmonic equations whose Dirichlet boundary values are gradient mappings.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Heinz‐type inequality and bi‐Lipschitz continuity for quasiconformal mappings satisfying inhomogeneous biharmonic equations

Zhong

Zhou

Yuan

2019

Math Methods in App Sciences

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Let Hom+false(double-struckTfalse) be the class of all sense‐preserving homeomorphic self‐mappings of double-struckT=false{z=x+iy∈double-struckC:false|zfalse|=1false}. The aim of this paper is twofold. First, we obtain Heinz‐type inequality for (K,K′)‐quasiconformal mappings satisfying inhomogeneous biharmonic equation Δ(Δω) = g in unit disk double-struckD with associated boundary value conditions normalΔω|T=φ∈scriptCfalse(double-struckTfalse) and ω|T=f∗∈Hom[+false(double-struckTfalse). Second, we establish biLipschitz continuity for (K,K′)‐quasiconformal mappings satisfying aforementioned inhomogeneous biharmonic equation when K′,false|false|φfalse||∞:=supz∈double-struckDfalse|φfalse(zfalse)false| and false|false|gfalse||∞:=supz∈double-struckDfalse|gfalse(zfalse)false| are small enough.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Heinz‐type inequality and bi‐Lipschitz continuity for quasiconformal mappings satisfying inhomogeneous biharmonic equations

Zhong

Zhou

Yuan

2019

Math Methods in App Sciences

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“…Then by [10,Inequality (3.6)], we have We will prove Φ is univalent in D r 0 , where r 0 satisfies the following equation: Thus, from the arbitrariness of z 1 and z 2 , the univalence of Φ follows. Now, we will prove Φ(D r 0 ) contains an univalent disk D R 0 To reach this goal, let ξ = r 0 e iθ ∈ ∂D r 0 .…”

Section: Schwarz-type Lemmas For Solutions To Inhomogeneous Biharmoni...mentioning

confidence: 96%

“…The main objective of this paper is to establish a Schwarz-type lemma for the solutions to the following inhomogeneous biharmonic Dirichlet problem (briefly, IBDP): We would like to mention that in [10] and [11], the authors have considered similar inhomogeneous biharmonic equations but with different boundaries conditions.…”

Section: Letmentioning

confidence: 99%

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Schwarz Lemma for mappings satisfying Biharmonic Equations

Khalfallah¹,

Haggui²,

Mhamdi³

2020

Preprint

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In this paper, we establish some Schwarz type lemmas for mappings Φ satisfying the inhomogeneous biharmonic Dirichlet problem ∆(∆(Φ)) = g in D, Φ = f on T and ∂ n Φ = h on T, where g is a continuous function on D, f, h are continuous functions on T, where D is the unit disc of the complex plane C and T = ∂D is the unit circle. To reach our aim, we start by investigating some properties of T 2 -harmonic functions. Finally, we prove a Landau-type theorem.

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“…It is well known that the Schwarz lemma has become a crucial theme in lots of branches of mathematical research for more than a hundred years to date. We refer the reader to [1,5,10,21,37,38,40,52,59,61] for more details on this topic. This paper continues the study of the classical Schwarz lemmas of holomorphic mappings and harmonic mappings (or complex-valued harmonic functions).…”

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confidence: 99%

Schwarz type lemmas and their applications in Banach spaces

Chen¹,

Hamada²,

Ponnusamy³

et al. 2021

Preprint

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The main purpose of this paper is to develop some methods to investigate the Schwarz type lemmas of holomorphic mappings and pluriharmonic mappings in Banach spaces. Initially, we extend the classical Schwarz lemmas of holomorphic mappings to Banach spaces, and then we apply these extensions to establish a sharp Bloch type theorem for pluriharmonic mappings on homogeneous unit balls of C n and to obtain some sharp boundary Schwarz type lemmas for holomorphic mappings in Banach spaces. Furthermore, we improve and generalize the classical Schwarz lemmas of planar harmonic mappings into the sharp forms of Banach spaces, and present some applications to sharp boundary Schwarz type lemmas for pluriharmonic mappings in Banach spaces. Additionally, using a relatively simple method of proof, we prove some sharp Schwarz-Pick type estimates of pluriharmonic mappings in JB * -triples, and the obtained results provide the improvements and generalizations of the corresponding results in [9].

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Schwarz-Type Lemma, Landau-Type Theorem, and Lipschitz-Type Space of Solutions to Inhomogeneous Biharmonic Equations

Cited by 18 publications

References 40 publications

Heinz‐type inequality and bi‐Lipschitz continuity for quasiconformal mappings satisfying inhomogeneous biharmonic equations

Heinz‐type inequality and bi‐Lipschitz continuity for quasiconformal mappings satisfying inhomogeneous biharmonic equations

Schwarz Lemma for mappings satisfying Biharmonic Equations

Schwarz type lemmas and their applications in Banach spaces

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