We investigate sharp upper bounds for the third-order Hankel determinants for two important subclasses ℱ and 𝒲 of normalized analytic functions f defined in an open unit disk satisfying Re f′ > 0, defined analytically by the conditions <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10986_2025_9670_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="110" /> </InlineMediaObject> <EquationSource Format="TEX">$\left|{f}{\prime}\left(z\right)-1\right|<1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfenced close="|" open="|"> <mi>f</mi> <mo>′</mo> <mfenced close=")" open="("> <mi>z</mi> </mfenced> <mo>-</mo> <mn>1</mn> </mfenced> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10986_2025_9670_Article_IEq2.gif" Format="GIF" Height="33" Rendition="HTML" Resolution="72" Type="Linedraw" Width="187" /> </InlineMediaObject> <EquationSource Format="TEX">$\text{Re}\left({f}{\prime}\left(z\right)+z{f}^{^{\prime\prime} }\left(z\right)\right)>0,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>Re</mtext> <mfenced close=")" open="("> <mi>f</mi> <mo>′</mo> <mfenced close=")" open="("> <mi>z</mi> </mfenced> <mo>+</mo> <mi>z</mi> <mmultiscripts> <mrow> <mi>f</mi> </mrow> <mrow /> <mmultiscripts> <mrow /> <mrow /> <mo>″</mo> </mmultiscripts> </mmultiscripts> <mfenced close=")" open="("> <mi>z</mi> </mfenced> </mfenced> <mo>></mo> <mn>0</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> respectively.

Hankel determinant for two subclasses of univalent functions with bounded turning

期刊论文
发表日期：

2025年4月5日

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Sumit Nagpal

摘要

We investigate sharp upper bounds for the third-order Hankel determinants for two important subclasses ℱ and 𝒲 of normalized analytic functions f defined in an open unit disk satisfying Re f′ > 0, defined analytically by the conditions $\left|{f}{\prime}\left(z\right)-1\right|<1$ $|f' (z) - 1| < 1$ and $\text{Re}\left({f}{\prime}\left(z\right)+z{f}^{^{\prime\prime} }\left(z\right)\right)>0,$ $Re (f' (z) + z f^{^{″}} (z)) > 0,$ respectively.