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<p>We prove the following convexity property of the function <Equation ID="Equ17"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2476_Article_Equ17.gif" Format="GIF" Height="43" Rendition="HTML" Resolution="72" Type="Linedraw" Width="228" /> </MediaObject> <EquationSource Format="TEX">\(\begin{aligned} R(c,x;t)= \frac{\sin t}{(\cos t)^c} \frac{x^c - (\cos t)^c}{x-\cos t}. \end{aligned}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi>R</mi> <mrow> <mo stretchy="false">(</mo> <mi>c</mi> <mo>,</mo> <mi>x</mi> <mo>;</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mo>sin</mo> <mi>t</mi> </mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mo>cos</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mi>c</mi> </msup> </mfrac> <mfrac> <mrow> <msup> <mi>x</mi> <mi>c</mi> </msup> <mo>-</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mo>cos</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mi>c</mi> </msup> </mrow> <mrow> <mi>x</mi> <mo>-</mo> <mo>cos</mo> <mi>t</mi> </mrow> </mfrac> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2476_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="77" /> </InlineMediaObject> <EquationSource Format="TEX">\(c\in \mathbb {R}{\setminus } \{0\}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>c</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> <mo lspace="0.15em" rspace="0.15em" stretchy="false">\</mo> <mo stretchy="false">{</mo> <mn>0</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>. The inequality <Equation ID="Equ18"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2476_Article_Equ18.gif" Format="GIF" Height="39" Rendition="HTML" Resolution="72" Type="Linedraw" Width="119" /> </MediaObject> <EquationSource Format="TEX">\(\begin{aligned} \frac{\partial ^2}{\partial t^2} R(c,x;t)\ge 0 \end{aligned}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mfrac> <msup> <mi>∂</mi> <mn>2</mn> </msup> <mrow> <mi>∂</mi> <msup> <mi>t</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mi>R</mi> <mrow> <mo stretchy="false">(</mo> <mi>c</mi> <mo>,</mo> <mi>x</mi> <mo>;</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>≥</mo> <mn>0</mn> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>holds for all <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2476_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="66" /> </InlineMediaObject> <EquationSource Format="TEX">\(x\in (0,1]\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2476_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="83" /> </InlineMediaObject> <EquationSource Format="TEX">\(t\in (0,\pi /2)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>π</mi> <mo stretchy="false">/</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> if and only if <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2476_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="153" /> </InlineMediaObject> <EquationSource Format="TEX">\(c\in [-2,-1]\cup (0,\infty )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>c</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mo>-</mo> <mn>2</mn> <mo>,</mo> <mo>-</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo>∪</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. This result settles a conjecture of M. Revers (1998).</p>

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Convexity and a Two-Parameter Class of Trigonometric Functions

  • 期刊论文

  • 发表日期：

    2025年7月15日

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  • Horst Alzer,
  • Man Kam Kwong

摘要

We prove the following convexity property of the function \(\begin{aligned} R(c,x;t)= \frac{\sin t}{(\cos t)^c} \frac{x^c - (\cos t)^c}{x-\cos t}. \end{aligned}\) R ( c , x ; t ) = sin t ( cos t ) c x c - ( cos t ) c x - cos t . Let \(c\in \mathbb {R}{\setminus } \{0\}\) c ∈ R \ { 0 } . The inequality \(\begin{aligned} \frac{\partial ^2}{\partial t^2} R(c,x;t)\ge 0 \end{aligned}\) ∂ 2 ∂ t 2 R ( c , x ; t ) ≥ 0 holds for all \(x\in (0,1]\) x ∈ ( 0 , 1 ] and \(t\in (0,\pi /2)\) t ∈ ( 0 , π / 2 ) if and only if \(c\in [-2,-1]\cup (0,\infty )\) c ∈ [ - 2 , - 1 ] ∪ ( 0 , ∞ ) . This result settles a conjecture of M. Revers (1998).