<p>Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1783_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="82" /> </InlineMediaObject> <EquationSource Format="TEX">$a\in (0, 1/2]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1783_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="66" /> </InlineMediaObject> <EquationSource Format="TEX">$r\in (0, 1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. In the article, we establish several sharp inequalities for generalized elliptic integral of the first kind <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1783_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {K}_{a}(r)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">K</mi> <mi>a</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, which refine and extend the previously well-known results of complete elliptic integral of the first kind <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1783_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="35" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {K}(r)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">K</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.</p>

Optimal bounds for generalized elliptic integral of the first kind

期刊论文
发表日期：

2025年8月15日

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Zai-Yin He

摘要

Let $a\in (0, 1/2]$ $a \in (0, 1 / 2]$ and $r\in (0, 1)$ $r \in (0, 1)$ . In the article, we establish several sharp inequalities for generalized elliptic integral of the first kind $\mathcal {K}_{a}(r)$ $K_{a} (r)$ , which refine and extend the previously well-known results of complete elliptic integral of the first kind $\mathcal {K}(r)$ $K (r)$ .