<p>For <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2111_Article_IEq3.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> we established the optimal estimate of the type <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2111_Article_IEq4.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="161" /> </InlineMediaObject> <EquationSource Format="TEX">$|Df(z)|\le \phi (|r|)\Vert f^{}\Vert _{p}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">|</mo> <mi>D</mi> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mrow> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mo stretchy="false">|</mo> <mi>r</mi> <mo stretchy="false">|</mo> <mo stretchy="false">)</mo> <mo stretchy="false">‖</mo> </mrow> <mmultiscripts> <mi>f</mi> <mrow /> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> <msub> <mrow> <mo stretchy="false">‖</mo> </mrow> <mi>p</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> where <i>f</i> is <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2111_Article_IEq1.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="28" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha -$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>-</mo> </mrow> </math></EquationSource> </InlineEquation>harmonic mapping defined in the unit disc belonging to the Hardy space <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2111_Article_IEq6.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">$h^{p},$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>h</mi> <mi>p</mi> </msup> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2111_Article_IEq7.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">$p\ge 1,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≥</mo> <mn>1</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2111_Article_IEq8.gif" Format="GIF" Height="25" Rendition="HTML" Resolution="72" Type="Linedraw" Width="72" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha +\frac{2}{p}\ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>+</mo> <mfrac> <mn>2</mn> <mi>p</mi> </mfrac> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2111_Article_IEq9.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$f^{}$</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mi>f</mi> <mrow /> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> </math></EquationSource> </InlineEquation> is its boundary function.</p>

Optimal Pointwise estimate for the derivative of $\alpha -$harmonic functions

期刊论文
发表日期：

2025年9月2日

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Djordjije Vujadinović,
Velimir Ćorović

摘要

For $\alpha >-1$ $α > - 1$ we established the optimal estimate of the type $|Df(z)|\le \phi (|r|)\Vert f^{*}\Vert _{p}$ $| D f (z) | \leq ϕ (| r |) ‖ f^{*} ‖_{p}$ where f is $\alpha -$ $α -$ harmonic mapping defined in the unit disc belonging to the Hardy space $h^{p},$ $h^{p},$ $p\ge 1,$ $p \geq 1,$ $\alpha +\frac{2}{p}\ge 0$ $α + \frac{2}{p} \geq 0$ and $f^{*}$ $f^{*}$ is its boundary function.

Optimal Pointwise estimate for the derivative of \(\alpha -\)harmonic functions

摘要