Abstract For 0 < q < 1, we define a new class <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11988_2025_5341_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">\({{\mathcal{R}}_{q}}\)</EquationSource>  </InlineEquation> of analytic functions using q-difference operator, which is a q-analogue of class <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11988_2025_5341_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal{R}\)</EquationSource>  </InlineEquation>. We prove that the class <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11988_2025_5341_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal{R}\)</EquationSource>  </InlineEquation> is properly contained in <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11988_2025_5341_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">\({{\mathcal{R}}_{q}}\)</EquationSource>  </InlineEquation> and find a sharp lower bound for the real part of <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11988_2025_5341_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="33" /> </InlineMediaObject> <EquationSource Format="TEX">\({{D}_{q}}f\)</EquationSource>  </InlineEquation>, where f ∈ <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11988_2025_5341_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">\({{\mathcal{R}}_{q}}\)</EquationSource>  </InlineEquation>. We investigate certain convolution properties of functions in the class <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11988_2025_5341_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">\({{\mathcal{R}}_{q}}\)</EquationSource>  </InlineEquation> including that the class <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11988_2025_5341_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">\({{\mathcal{R}}_{q}}\)</EquationSource>  </InlineEquation> is closed under convolution for a definite range of q. The findings of the present manuscript essentially generalizes some well-known results in the literature.

A q-Generalization of Class \(\mathcal{R}\)

期刊论文
发表日期：

2025年6月1日

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Sarika Verma,
Raj Kumar,
Janusz Sokół,
Sukhjit Singh

摘要

Abstract

For 0 < q < 1, we define a new class \({{\mathcal{R}}_{q}}\) of analytic functions using q-difference operator, which is a q-analogue of class \(\mathcal{R}\) . We prove that the class \(\mathcal{R}\) is properly contained in \({{\mathcal{R}}_{q}}\) and find a sharp lower bound for the real part of \({{D}_{q}}f\) , where f ∈ \({{\mathcal{R}}_{q}}\) . We investigate certain convolution properties of functions in the class \({{\mathcal{R}}_{q}}\) including that the class \({{\mathcal{R}}_{q}}\) is closed under convolution for a definite range of q. The findings of the present manuscript essentially generalizes some well-known results in the literature.