{"title":"The Heinz type inequality, Bloch type theorem and Lipschitz characteristic of polyharmonic mappings","authors":"Shaolin Chen","doi":"10.1016/j.indag.2023.05.001","DOIUrl":null,"url":null,"abstract":"<div><p>Suppose that <span><math><mi>f</mi></math></span> satisfies the following: <span><math><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></math></span> the polyharmonic equation <span><math><mrow><msup><mrow><mi>Δ</mi></mrow><mrow><mi>m</mi></mrow></msup><mi>f</mi><mo>=</mo><mi>Δ</mi><mrow><mo>(</mo><msup><mrow><mi>Δ</mi></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>f</mi><mo>)</mo></mrow></mrow></math></span>\n<span><math><mrow><mo>=</mo><msub><mrow><mi>φ</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></math></span>\n<span><math><mrow><mo>(</mo><msub><mrow><mi>φ</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>∈</mo><mi>C</mi><mrow><mo>(</mo><mover><mrow><msup><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow><mo>¯</mo></mover><mo>,</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></mrow><mo>)</mo></mrow></math></span>, (2) the boundary conditions <span><math><mrow><msup><mrow><mi>Δ</mi></mrow><mrow><mn>0</mn></mrow></msup><mi>f</mi><mo>=</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><msup><mrow><mi>Δ</mi></mrow><mrow><mn>1</mn></mrow></msup><mi>f</mi><mo>=</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mspace></mspace><mo>…</mo><mo>,</mo><mspace></mspace><msup><mrow><mi>Δ</mi></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>f</mi><mo>=</mo><msub><mrow><mi>φ</mi></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></math></span> on <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span>\n(<span><math><mrow><msub><mrow><mi>φ</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>∈</mo><mi>C</mi><mrow><mo>(</mo><msup><mrow><mi>S</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>,</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span> for <span><math><mrow><mi>j</mi><mo>∈</mo><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>m</mi><mo>−</mo><mn>1</mn><mo>}</mo></mrow></mrow></math></span> and <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> denotes the boundary of the unit ball <span><math><msup><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>), and <span><math><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></math></span>\n<span><math><mrow><mi>f</mi><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></math></span>, where <span><math><mrow><mi>n</mi><mo>≥</mo><mn>3</mn></mrow></math></span> and <span><math><mrow><mi>m</mi><mo>≥</mo><mn>1</mn></mrow></math></span> are integers. Initially, we prove a Schwarz type lemma and use it to obtain a Heinz type inequality of mappings satisfying the polyharmonic equation with the above Dirichlet boundary value conditions. Furthermore, we establish a Bloch type theorem of mappings satisfying the above polyharmonic equation, which gives an answer to an open problem in Chen and Ponnusamy, (2019). Additionally, we show that if <span><math><mi>f</mi></math></span> is a <span><math><mi>K</mi></math></span>-quasiconformal self-mapping of <span><math><msup><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> satisfying the above polyharmonic equation, then <span><math><mi>f</mi></math></span><span> is Lipschitz continuous, and the Lipschitz constant is asymptotically sharp as </span><span><math><mrow><mi>K</mi><mo>→</mo><msup><mrow><mn>1</mn></mrow><mrow><mo>+</mo></mrow></msup></mrow></math></span> and <span><math><mrow><msub><mrow><mo>‖</mo><msub><mrow><mi>φ</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>‖</mo></mrow><mrow><mi>∞</mi></mrow></msub><mo>→</mo><msup><mrow><mn>0</mn></mrow><mrow><mo>+</mo></mrow></msup></mrow></math></span> for <span><math><mrow><mi>j</mi><mo>∈</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>m</mi><mo>}</mo></mrow></mrow></math></span>.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":null,"pages":null},"PeriodicalIF":0.5000,"publicationDate":"2023-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Indagationes Mathematicae-New Series","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0019357723000411","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 1
Abstract
Suppose that satisfies the following: the polyharmonic equation
, (2) the boundary conditions on
( for and denotes the boundary of the unit ball ), and
, where and are integers. Initially, we prove a Schwarz type lemma and use it to obtain a Heinz type inequality of mappings satisfying the polyharmonic equation with the above Dirichlet boundary value conditions. Furthermore, we establish a Bloch type theorem of mappings satisfying the above polyharmonic equation, which gives an answer to an open problem in Chen and Ponnusamy, (2019). Additionally, we show that if is a -quasiconformal self-mapping of satisfying the above polyharmonic equation, then is Lipschitz continuous, and the Lipschitz constant is asymptotically sharp as and for .
期刊介绍:
Indagationes Mathematicae is a peer-reviewed international journal for the Mathematical Sciences of the Royal Dutch Mathematical Society. The journal aims at the publication of original mathematical research papers of high quality and of interest to a large segment of the mathematics community. The journal also welcomes the submission of review papers of high quality.