Generalized harmonic functions and Schwarz lemma for biharmonic mappings

Adel Khalfallah; Fathi Haggui; Mohamed Mhamdi

doi:10.1007/s00605-021-01619-4

Generalized harmonic functions and Schwarz lemma for biharmonic mappings

Adel Khalfallah^*, Fathi Haggui, Mohamed Mhamdi

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

6 Scopus citations

Abstract

In this paper, we establish some Schwarz type lemmas for mappings Φ satisfying the inhomogeneous biharmonic Dirichlet problem Δ (Δ (Φ)) = g in D, Φ = f on T and ∂_nΦ = 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 generalized harmonic functions called T_α-harmonic functions. Finally, we prove a Landau-type theorem for this class of functions, when α> 0.

Original language	English
Pages (from-to)	823-849
Number of pages	27
Journal	Monatshefte fur Mathematik
Volume	196
Issue number	4
DOIs	https://doi.org/10.1007/s00605-021-01619-4
State	Published - Dec 2021

Bibliographical note

Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Austria, part of Springer Nature.

Keywords

Biharmonic equations
Boundary Schwarz’s lemma
Landau theorem
Schwarz’s lemma
T-harmonic mappings

ASJC Scopus subject areas

General Mathematics

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10.1007/s00605-021-01619-4

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title = "Generalized harmonic functions and Schwarz lemma for biharmonic mappings",

abstract = "In this paper, we establish some Schwarz type lemmas for mappings Φ satisfying the inhomogeneous biharmonic Dirichlet problem Δ (Δ (Φ)) = g in D, Φ = f on T and ∂nΦ = 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 generalized harmonic functions called Tα-harmonic functions. Finally, we prove a Landau-type theorem for this class of functions, when α> 0.",

keywords = "Biharmonic equations, Boundary Schwarz{\textquoteright}s lemma, Landau theorem, Schwarz{\textquoteright}s lemma, T-harmonic mappings",

author = "Adel Khalfallah and Fathi Haggui and Mohamed Mhamdi",

note = "Publisher Copyright: {\textcopyright} 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Austria, part of Springer Nature.",

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month = dec,

doi = "10.1007/s00605-021-01619-4",

language = "English",

volume = "196",

pages = "823--849",

journal = "Monatshefte fur Mathematik",

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TY - JOUR

T1 - Generalized harmonic functions and Schwarz lemma for biharmonic mappings

AU - Khalfallah, Adel

AU - Haggui, Fathi

AU - Mhamdi, Mohamed

PY - 2021/12

Y1 - 2021/12

N2 - In this paper, we establish some Schwarz type lemmas for mappings Φ satisfying the inhomogeneous biharmonic Dirichlet problem Δ (Δ (Φ)) = g in D, Φ = f on T and ∂nΦ = 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 generalized harmonic functions called Tα-harmonic functions. Finally, we prove a Landau-type theorem for this class of functions, when α> 0.

AB - In this paper, we establish some Schwarz type lemmas for mappings Φ satisfying the inhomogeneous biharmonic Dirichlet problem Δ (Δ (Φ)) = g in D, Φ = f on T and ∂nΦ = 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 generalized harmonic functions called Tα-harmonic functions. Finally, we prove a Landau-type theorem for this class of functions, when α> 0.

KW - Biharmonic equations

KW - Boundary Schwarz’s lemma

KW - Landau theorem

KW - Schwarz’s lemma

KW - T-harmonic mappings

UR - https://www.scopus.com/pages/publications/85113826379

U2 - 10.1007/s00605-021-01619-4

DO - 10.1007/s00605-021-01619-4

M3 - Article

AN - SCOPUS:85113826379

SN - 0026-9255

VL - 196

SP - 823

EP - 849

JO - Monatshefte fur Mathematik

JF - Monatshefte fur Mathematik

IS - 4

ER -

Generalized harmonic functions and Schwarz lemma for biharmonic mappings

Abstract

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Keywords

ASJC Scopus subject areas

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