<p>For <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_63_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="78" /> </InlineMediaObject> <EquationSource Format="TEX">\(\alpha \in (0,\infty)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, let <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_63_Article_IEq5.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal{B}_{\mathcal{H},\Omega}(\alpha)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">B</mi> <mrow> <mi mathvariant="script">H</mi> <mo>,</mo> <mi mathvariant="normal">Ω</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> denote the class of <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_63_Article_IEq6.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>-Bloch mappings on a proper simply connected domain <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_63_Article_IEq7.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Omega \subseteq \mathbb{C}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊆</mo> <mi mathvariant="double-struck">C</mi> </mrow> </math></EquationSource> </InlineEquation>. In this article, we introduce the class <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_63_Article_IEq8.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal{B}^{*}_{\mathcal{H},\Omega}(\alpha)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mrow> <mi mathvariant="script">B</mi> </mrow> <mrow> <mi mathvariant="script">H</mi> <mo>,</mo> <mi mathvariant="normal">Ω</mi> </mrow> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of harmonic <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_63_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>-Bloch-type mappings on a proper simply connected domain <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_63_Article_IEq10.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Omega \subseteq \mathbb{C}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊆</mo> <mi mathvariant="double-struck">C</mi> </mrow> </math></EquationSource> </InlineEquation> and study several interesting properties of the classes <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_63_Article_IEq11.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal{B}_{\mathcal{H},\Omega}(\alpha)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">B</mi> <mrow> <mi mathvariant="script">H</mi> <mo>,</mo> <mi mathvariant="normal">Ω</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_63_Article_IEq12.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal{B}^{*}_{\mathcal{H},\Omega}(\alpha)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mrow> <mi mathvariant="script">B</mi> </mrow> <mrow> <mi mathvariant="script">H</mi> <mo>,</mo> <mi mathvariant="normal">Ω</mi> </mrow> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> when <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_63_Article_IEq13.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Omega\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> is proper simply connected domain and the shifted disk <InlineEquation ID="IEq14"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_63_Article_IEq14.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Omega_{\gamma}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Ω</mi> <mi>γ</mi> </msub> </math></EquationSource> </InlineEquation> containing <InlineEquation ID="IEq15"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_63_Article_IEq15.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 <Equation ID="Equa"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_63_Article_Equa.gif" Format="GIF" Height="40" Rendition="HTML" Resolution="72" Type="Linedraw" Width="266" /> </MediaObject> <EquationSource Format="TEX">\(\Omega_{\gamma}:=\big\{z\in\mathbb{C} : \big|z+\frac{\gamma}{1-\gamma}\big|&lt;\frac{1}{1-\gamma}\big\}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msub> <mi mathvariant="normal">Ω</mi> <mi>γ</mi> </msub> <mo>:</mo> <mo>=</mo> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">{</mo> </mrow> <mi>z</mi> <mo>∈</mo> <mi mathvariant="double-struck">C</mi> <mo>:</mo> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">|</mo> </mrow> <mi>z</mi> <mo>+</mo> <mfrac> <mi>γ</mi> <mrow> <mn>1</mn> <mo>-</mo> <mi>γ</mi> </mrow> </mfrac> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">|</mo> </mrow> <mo>&lt;</mo> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>-</mo> <mi>γ</mi> </mrow> </mfrac> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">}</mo> </mrow> </mrow> </math></EquationSource> </Equation> and <InlineEquation ID="IEq16"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_63_Article_IEq16.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="73" /> </InlineMediaObject> <EquationSource Format="TEX">\(0 \leq \gamma &lt;1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>≤</mo> <mi>γ</mi> <mo>&lt;</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. For <InlineEquation ID="IEq17"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_63_Article_IEq17.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="91" /> </InlineMediaObject> <EquationSource Format="TEX">\(f \in \mathcal{B}_{\mathcal{H},\Omega}(\alpha)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>∈</mo> <msub> <mi mathvariant="script">B</mi> <mrow> <mi mathvariant="script">H</mi> <mo>,</mo> <mi mathvariant="normal">Ω</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> (respectively <InlineEquation ID="IEq18"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_63_Article_IEq18.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="65" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal{B}^{*}_{\mathcal{H},\Omega}(\alpha))\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mrow> <mi mathvariant="script">B</mi> </mrow> <mrow> <mi mathvariant="script">H</mi> <mo>,</mo> <mi mathvariant="normal">Ω</mi> </mrow> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> <mrow> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of the form <InlineEquation ID="IEq19"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_63_Article_IEq19.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="329" /> </InlineMediaObject> <EquationSource Format="TEX">\(f(z)=h(z) + \overline{g(z)}=\sum_{n=0}^{\infty}a_nz^n + \overline{\sum_{n=1}^{\infty}b_nz^n}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>h</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mover> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>¯</mo> </mover> <mo>=</mo> <msubsup> <mo>∑</mo> <mrow> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>∞</mi> </msubsup> <msub> <mi>a</mi> <mi>n</mi> </msub> <msup> <mi>z</mi> <mi>n</mi> </msup> <mo>+</mo> <mover> <mrow> <msubsup> <mo>∑</mo> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>∞</mi> </msubsup> <msub> <mi>b</mi> <mi>n</mi> </msub> <msup> <mi>z</mi> <mi>n</mi> </msup> </mrow> <mo>¯</mo> </mover> </mrow> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq20"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_63_Article_IEq20.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 Bloch norm <InlineEquation ID="IEq21"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_63_Article_IEq21.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="94" /> </InlineMediaObject> <EquationSource Format="TEX">\( \lVert f \rVert _{\mathcal{H},\Omega, \alpha} \leq 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mrow> <mo stretchy="false">‖</mo> <mi>f</mi> <mo stretchy="false">‖</mo> </mrow> <mrow> <mi mathvariant="script">H</mi> <mo>,</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mi>α</mi> </mrow> </msub> <mo>≤</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> (respectively <InlineEquation ID="IEq22"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_63_Article_IEq22.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="94" /> </InlineMediaObject> <EquationSource Format="TEX">\( \lVert f \rVert ^{*}_{\mathcal{H},\Omega, \alpha} \leq 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mrow> <mo stretchy="false">‖</mo> <mi>f</mi> <mo stretchy="false">‖</mo> </mrow> <mrow> <mi mathvariant="script">H</mi> <mo>,</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mi>α</mi> </mrow> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> <mo>≤</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>), we define the Bloch–Bohr radius for the class <InlineEquation ID="IEq23"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_63_Article_IEq23.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal{B}_{\mathcal{H},\Omega}(\alpha)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">B</mi> <mrow> <mi mathvariant="script">H</mi> <mo>,</mo> <mi mathvariant="normal">Ω</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> (respectively <InlineEquation ID="IEq24"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_63_Article_IEq24.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="65" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal{B}^{*}_{\mathcal{H},\Omega}(\alpha))\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mrow> <mi mathvariant="script">B</mi> </mrow> <mrow> <mi mathvariant="script">H</mi> <mo>,</mo> <mi mathvariant="normal">Ω</mi> </mrow> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> <mrow> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> to be the largest radius <InlineEquation ID="IEq25"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_63_Article_IEq25.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="86" /> </InlineMediaObject> <EquationSource Format="TEX">\(r_{\Omega,\alpha} \in (0,1)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>r</mi> <mrow> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mi>α</mi> </mrow> </msub> <mo>∈</mo> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq26"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_63_Article_IEq26.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="171" /> </InlineMediaObject> <EquationSource Format="TEX">\(\sum_{n=0}^{\infty}(|a_n|+|b_{n}|) r^n\leq 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mo>∑</mo> <mrow> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>∞</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">|</mo> <mo>+</mo> <mo stretchy="false">|</mo> </mrow> <msub> <mi>b</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">|</mo> <mo stretchy="false">)</mo> </mrow> <msup> <mi>r</mi> <mi>n</mi> </msup> <mo>≤</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq27"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_63_Article_IEq27.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="61" /> </InlineMediaObject> <EquationSource Format="TEX">\(r \leq r_{\Omega, \alpha}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>≤</mo> <msub> <mi>r</mi> <mrow> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mi>α</mi> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation> and for all <InlineEquation ID="IEq28"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_63_Article_IEq28.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="91" /> </InlineMediaObject> <EquationSource Format="TEX">\(f \in \mathcal{B}_{\mathcal{H},\Omega}(\alpha)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>∈</mo> <msub> <mi mathvariant="script">B</mi> <mrow> <mi mathvariant="script">H</mi> <mo>,</mo> <mi mathvariant="normal">Ω</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> (respectively <InlineEquation ID="IEq29"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_63_Article_IEq29.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="65" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal{B}^{*}_{\mathcal{H},\Omega}(\alpha))\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mrow> <mi mathvariant="script">B</mi> </mrow> <mrow> <mi mathvariant="script">H</mi> <mo>,</mo> <mi mathvariant="normal">Ω</mi> </mrow> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> <mrow> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We also investigate Bloch–Bohr radius for the classes <InlineEquation ID="IEq30"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_63_Article_IEq30.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal{B}_{\mathcal{H},\Omega}(\alpha)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">B</mi> <mrow> <mi mathvariant="script">H</mi> <mo>,</mo> <mi mathvariant="normal">Ω</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq31"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_63_Article_IEq31.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal{B}^{*}_{\mathcal{H},\Omega}(\alpha)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mrow> <mi mathvariant="script">B</mi> </mrow> <mrow> <mi mathvariant="script">H</mi> <mo>,</mo> <mi mathvariant="normal">Ω</mi> </mrow> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> on simply connected domain <InlineEquation ID="IEq32"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_63_Article_IEq32.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Omega\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> containing <InlineEquation ID="IEq322"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_63_Article_IEq322.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>.</p>