<p>In this paper, we study Riesz-Fejér inequality, comparative growth of integral means and boundary behavior for solutions of the <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_71_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha$</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-harmonic equation in the unit disk <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_71_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb{D}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">D</mi> </math></EquationSource> </InlineEquation>. For <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_71_Article_IEq4.gif" Format="GIF" Height="25" Rendition="HTML" Resolution="72" Type="Linedraw" Width="136" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha>\max\{-1,-\frac{2}{p}\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>></mo> <mo movablelimits="true">max</mo> <mo stretchy="false">{</mo> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mo>-</mo> <mfrac> <mn>2</mn> <mi>p</mi> </mfrac> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_71_Article_IEq5.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha \geq 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_71_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="83" /> </InlineMediaObject> <EquationSource Format="TEX">$1<p<\infty$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, we obtain a Riesz-Fejér inequality for functions in the real kernel <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_71_Article_IEq7.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha$</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-harmonic Hardy space consisting of solutions <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_71_Article_IEq8.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$u$</EquationSource> <EquationSource Format="MATHML"><math> <mi>u</mi> </math></EquationSource> </InlineEquation> of the <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_71_Article_IEq9.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha$</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-harmonic equation in <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_71_Article_IEq10.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb{D}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">D</mi> </math></EquationSource> </InlineEquation> with uniformly bounded integral mean <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_71_Article_IEq11.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="61" /> </InlineMediaObject> <EquationSource Format="TEX">$M_{p}(r, u)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>M</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with respect to <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_71_Article_IEq12.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="66" /> </InlineMediaObject> <EquationSource Format="TEX">$r\in(0,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Furthermore, for <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_71_Article_IEq13.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="114" /> </InlineMediaObject> <EquationSource Format="TEX">$1\leq p<q\leq\infty$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo><</mo> <mi>q</mi> <mo>≤</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, we estimate the growth of <InlineEquation ID="IEq14"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_71_Article_IEq14.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="61" /> </InlineMediaObject> <EquationSource Format="TEX">$M_{q}(r,u)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>M</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> if the growth of <InlineEquation ID="IEq15"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_71_Article_IEq15.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="61" /> </InlineMediaObject> <EquationSource Format="TEX">$M_{p}(r,u)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>M</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is known. Moreover, we consider the boundary behavior of real kernel <InlineEquation ID="IEq16"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_71_Article_IEq16.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha$</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-Poisson integrals in <InlineEquation ID="IEq17"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_71_Article_IEq17.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb{D}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">D</mi> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq18"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_71_Article_IEq18.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha>-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>></mo> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. Our results generalize the related previous results.</p>

Properties of solutions of the $\alpha$-harmonic equation in the unit disk

期刊论文
发表日期：

2025年2月24日

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Z. Y. Hu,
J. H. Fan,
H. M. Srivastava

摘要

In this paper, we study Riesz-Fejér inequality, comparative growth of integral means and boundary behavior for solutions of the $\alpha$ $α$ -harmonic equation in the unit disk $\mathbb{D}$ $D$ . For $\alpha>\max\{-1,-\frac{2}{p}\}$ $α > max {- 1, - \frac{2}{p}}$ $\alpha \geq 0$ $α \geq 0$ and $1<p<\infty$ $1 < p < \infty$ , we obtain a Riesz-Fejér inequality for functions in the real kernel $\alpha$ $α$ -harmonic Hardy space consisting of solutions $u$ $u$ of the $\alpha$ $α$ -harmonic equation in $\mathbb{D}$ $D$ with uniformly bounded integral mean $M_{p}(r, u)$ $M_{p} (r, u)$ with respect to $r\in(0,1)$ $r \in (0, 1)$ . Furthermore, for $1\leq p<q\leq\infty$ $1 \leq p < q \leq \infty$ , we estimate the growth of $M_{q}(r,u)$ $M_{q} (r, u)$ if the growth of $M_{p}(r,u)$ $M_{p} (r, u)$ is known. Moreover, we consider the boundary behavior of real kernel $\alpha$ $α$ -Poisson integrals in $\mathbb{D}$ $D$ , where $\alpha>-1$ $α > - 1$ . Our results generalize the related previous results.

Properties of solutions of the \(\alpha\)-harmonic equation in the unit disk

摘要