Researchers have been greatly impacted by the recent connections between geometric function theory and certain special polynomials of certain subclasses in geometric function theory made use of the Chebyshev, Faber, Horadam, Lucas, and Fibonacci polynomials and their generalizations. Many subclasses of analytic functions were examined by considering the well-known subordination notion and scientific understanding in geometric function theory. In this work, we present a novel subfamily of analytic and bi-univalent functions in connection with Second Einstein Function ( SEF), drawing inspiration from the relationships between special polynomials and analytic function classes. First time we define a new class <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1848_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="64" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathfrak {B}_{\Sigma }^\wp (\tau ,\kappa )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi mathvariant="fraktur">B</mi> <mrow> <mi mathvariant="normal">Σ</mi> </mrow> <mi>℘</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>,</mo> <mi>κ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of Babalola- type bi-starlike functions based on Erdély–Kober Fractional-Order Derivative subordinating with SEF. We determine the bounds of initial Taylor coefficients <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1848_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">$|a_2|$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">|</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1848_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">$|a_3|$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mn>3</mn> </msub> <mrow> <mo stretchy="false">|</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for functions in <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1848_Article_IEq4.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="69" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathfrak {B}_{\Sigma }^\wp (\tau ,\kappa ).$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi mathvariant="fraktur">B</mi> <mrow> <mi mathvariant="normal">Σ</mi> </mrow> <mi>℘</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>,</mo> <mi>κ</mi> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> Besides, using the values of <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1848_Article_IEq5.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$a_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>a</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1848_Article_IEq6.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$a_3$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>a</mi> <mn>3</mn> </msub> </math></EquationSource> </InlineEquation> we investigate the Fekete-Szegő functional for <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1848_Article_IEq7.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="101" /> </InlineMediaObject> <EquationSource Format="TEX">$f\in \mathfrak {B}_{\Sigma }^\wp (\tau ,\kappa ).$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>∈</mo> <msubsup> <mi mathvariant="fraktur">B</mi> <mrow> <mi mathvariant="normal">Σ</mi> </mrow> <mi>℘</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>,</mo> <mi>κ</mi> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> By specialising the parameters <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1848_Article_IEq8.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="30" /> </InlineMediaObject> <EquationSource Format="TEX">$\wp ,\tau $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>℘</mi> <mo>,</mo> <mi>τ</mi> </mrow> </math></EquationSource> </InlineEquation>, we deduce various new subclasses of <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1848_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="64" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathfrak {B}_{\Sigma }^\wp (\tau ,\kappa )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi mathvariant="fraktur">B</mi> <mrow> <mi mathvariant="normal">Σ</mi> </mrow> <mi>℘</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>,</mo> <mi>κ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and some attractive interpretation of our results which have not been discussed so far are pointed out as corollaries.

On Babalola-Type Bi-Starlike Functions Based on Erdély–Kober Fractional-Order Derivative Subordinate to Second Einstein Function

期刊论文
发表日期：

2025年10月28日

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G. Murugusundaramoorthy,
K. Vijaya,
Luminiţa-Ioana Cotîrlǎ

摘要

Researchers have been greatly impacted by the recent connections between geometric function theory and certain special polynomials of certain subclasses in geometric function theory made use of the Chebyshev, Faber, Horadam, Lucas, and Fibonacci polynomials and their generalizations. Many subclasses of analytic functions were examined by considering the well-known subordination notion and scientific understanding in geometric function theory. In this work, we present a novel subfamily of analytic and bi-univalent functions in connection with Second Einstein Function ( SEF), drawing inspiration from the relationships between special polynomials and analytic function classes. First time we define a new class $\mathfrak {B}_{\Sigma }^\wp (\tau ,\kappa )$ $B_{Σ}^{℘} (τ, κ)$ of Babalola- type bi-starlike functions based on Erdély–Kober Fractional-Order Derivative subordinating with SEF. We determine the bounds of initial Taylor coefficients $|a_2|$ $| a_{2} |$ and $|a_3|$ $| a_{3} |$ for functions in $\mathfrak {B}_{\Sigma }^\wp (\tau ,\kappa ).$ $B_{Σ}^{℘} (τ, κ) .$ Besides, using the values of $a_2$ $a_{2}$ and $a_3$ $a_{3}$ we investigate the Fekete-Szegő functional for $f\in \mathfrak {B}_{\Sigma }^\wp (\tau ,\kappa ).$ $f \in B_{Σ}^{℘} (τ, κ) .$ By specialising the parameters $\wp ,\tau $ $℘, τ$ , we deduce various new subclasses of $\mathfrak {B}_{\Sigma }^\wp (\tau ,\kappa )$ $B_{Σ}^{℘} (τ, κ)$ and some attractive interpretation of our results which have not been discussed so far are pointed out as corollaries.