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<p>We establish sharp inequalities involving the incomplete Beta and Gamma functions. These inequalities arise in the approximation of generalized Bernstein functions by higher order Thorin-Bernstein functions. Furthermore, new properties of a related function, namely <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(x^{\lambda }\Gamma (x)/\Gamma (x+\lambda )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>x</mi> <mi>λ</mi> </msup> <mi mathvariant="normal">Γ</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">/</mo> <mi mathvariant="normal">Γ</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> are derived.</p>

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Approximations of Generalized Bernstein Functions

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年3月25日

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  • Stamatis Koumandos,
  • Henrik L. Pedersen

摘要

We establish sharp inequalities involving the incomplete Beta and Gamma functions. These inequalities arise in the approximation of generalized Bernstein functions by higher order Thorin-Bernstein functions. Furthermore, new properties of a related function, namely \(x^{\lambda }\Gamma (x)/\Gamma (x+\lambda )\) x λ Γ ( x ) / Γ ( x + λ ) are derived.