<p>By using beta-negative binomial distribution, we introduce two novel subclasses of spiral-like functions; namely, spiral-starlike functions and spiral-convex functions denoted by <InlineEquation ID="IEq1"> <EquationSource Format="TEX">${S}_{\lambda ,\gamma ,\mu }^{\eta }\left(\alpha ,\beta \right)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>S</mi> <mrow> <mi>λ</mi> <mo>,</mo> <mi>γ</mi> <mo>,</mo> <mi>μ</mi> </mrow> <mi>η</mi> </msubsup> <mfenced close=")" open="("> <mi>α</mi> <mo>,</mo> <mi>β</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">${E}_{\lambda ,\gamma ,\mu }^{\eta }\left(\alpha ,\beta \right)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>E</mi> <mrow> <mi>λ</mi> <mo>,</mo> <mi>γ</mi> <mo>,</mo> <mi>μ</mi> </mrow> <mi>η</mi> </msubsup> <mfenced close=")" open="("> <mi>α</mi> <mo>,</mo> <mi>β</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation> respectively, and defined in the domain of open unit disk <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${\mathbb{D}}=\left\{z\in {\mathbb{C}}:\left|z\right|<1\right\}.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">D</mi> <mo>=</mo> <mfenced close="}" open="{"> <mi>z</mi> <mo>∈</mo> <mi mathvariant="double-struck">C</mi> <mo>:</mo> <mfenced close="|" open="|"> <mi>z</mi> </mfenced> <mo><</mo> <mn>1</mn> </mfenced> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> We establish sufficient conditions for functions to be members of families mentioned above. Further, the bounds of some initial coefficients and Fekete–Szegö functionals for the classes described above are obtained.</p>

Some Properties of Subclasses of Spiral-Like Functions Involving Beta-Negative Binomial Distribution

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发表日期：

2026年6月3日

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Eureka Pattnayak,
Trailokya Panigrahi,
Rabha M. El-Ashwah

摘要

By using beta-negative binomial distribution, we introduce two novel subclasses of spiral-like functions; namely, spiral-starlike functions and spiral-convex functions denoted by ${S}_{\lambda ,\gamma ,\mu }^{\eta }\left(\alpha ,\beta \right)$ $S_{λ, γ, μ}^{η} (α, β)$ and ${E}_{\lambda ,\gamma ,\mu }^{\eta }\left(\alpha ,\beta \right)$ $E_{λ, γ, μ}^{η} (α, β)$ respectively, and defined in the domain of open unit disk ${\mathbb{D}}=\left\{z\in {\mathbb{C}}:\left|z\right|<1\right\}.$ $D = \{z \in C : |z| < 1\} .$ We establish sufficient conditions for functions to be members of families mentioned above. Further, the bounds of some initial coefficients and Fekete–Szegö functionals for the classes described above are obtained.