<p>In this paper, we consider the class <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1888_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {S}}^(\alpha )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="script">S</mi> </mrow> <mo>∗</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of starlike functions of order <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1888_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1888_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="67" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha \in [0,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. We find the sharp bound of the third Hankel determinant for the inverse function <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1888_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="28" /> </InlineMediaObject> <EquationSource Format="TEX">$f^{-1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>f</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> when <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1888_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="75" /> </InlineMediaObject> <EquationSource Format="TEX">$f \in {\mathcal {S}}^(\alpha )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="script">S</mi> </mrow> <mo>∗</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1888_Article_IEq8.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="78" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha \in \left[ \frac{3}{8}, \frac{17}{18} \right] $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mfenced close="]" open="["> <mfrac> <mn>3</mn> <mn>8</mn> </mfrac> <mo>,</mo> <mfrac> <mn>17</mn> <mn>18</mn> </mfrac> </mfenced> </mrow> </math></EquationSource> </InlineEquation>. This result is a generalization of the up-to-date known bounds of <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1888_Article_IEq9.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="69" /> </InlineMediaObject> <EquationSource Format="TEX">$H_{3,1}(f^{-1})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>H</mi> <mrow> <mn>3</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> when <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1888_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="75" /> </InlineMediaObject> <EquationSource Format="TEX">$f \in {\mathcal {S}}^*(\alpha )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="script">S</mi> </mrow> <mo>∗</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1888_Article_IEq11.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha =0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1888_Article_IEq12.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha =\frac{1}{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </math></EquationSource> </InlineEquation>.</p>

Third Hankel Determinant for the Inverse of Starlike Functions of Order $\alpha $

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2025年5月22日

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Katarzyna Tra̧bka-Wiȩcław,
Paweł Zaprawa

摘要

In this paper, we consider the class ${\mathcal {S}}^*(\alpha )$ $S^{*} (α)$ of starlike functions of order $\alpha $ $α$ , $\alpha \in [0,1)$ $α \in [0, 1)$ . We find the sharp bound of the third Hankel determinant for the inverse function $f^{-1}$ $f^{- 1}$ when $f \in {\mathcal {S}}^*(\alpha )$ $f \in S^{*} (α)$ and $\alpha \in \left[ \frac{3}{8}, \frac{17}{18} \right] $ $α \in [\frac{3}{8}, \frac{17}{18}]$ . This result is a generalization of the up-to-date known bounds of $H_{3,1}(f^{-1})$ $H_{3, 1} (f^{- 1})$ when $f \in {\mathcal {S}}^*(\alpha )$ $f \in S^{*} (α)$ for $\alpha =0$ $α = 0$ and $\alpha =\frac{1}{2}$ $α = \frac{1}{2}$ .

Third Hankel Determinant for the Inverse of Starlike Functions of Order \(\alpha \)

摘要