<p>We prove that the function <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_69_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="155" /> </InlineMediaObject> <EquationSource Format="TEX">$g(x)= 1 / ( 1 - \cos(x) )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <mo>cos</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is completely monotonic on <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_69_Article_IEq11.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="38" /> </InlineMediaObject> <EquationSource Format="TEX">$(0,\pi]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>π</mi> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> and absolutely monotonic on <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_69_Article_IEq12.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$[\pi, 2\pi)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mi>π</mi> <mo>,</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, and we determine the best possible bounds <InlineEquation ID="IEq21"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_69_Article_IEq21.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>λ</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq22"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_69_Article_IEq22.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$\mu_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>μ</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> such that the inequalities<Equation ID="Equa"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_69_Article_Equa.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="358" /> </MediaObject> <EquationSource Format="TEX">$\lambda_n \leq g^{(n)}(x)+g^{(n)}(y)-g^{(n)}(x+y) \quad (n \geq 0 \ \mbox{even})$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msub> <mi>λ</mi> <mi>n</mi> </msub> <mo>≤</mo> <msup> <mi>g</mi> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <msup> <mi>g</mi> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <msup> <mi>g</mi> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="1em" /> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>≥</mo> <mn>0</mn> <mspace width="4pt" /> <mtext>even</mtext> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </Equation>and<Equation ID="Equb"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_69_Article_Equb.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="356" /> </MediaObject> <EquationSource Format="TEX">$\mu_n \leq g^{(n)}(x+y)-g^{(n)}(x)-g^{(n)}(y) \quad (n \geq 1 \ \mbox{odd})$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msub> <mi>μ</mi> <mi>n</mi> </msub> <mo>≤</mo> <msup> <mi>g</mi> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <msup> <mi>g</mi> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <msup> <mi>g</mi> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="1em" /> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>≥</mo> <mn>1</mn> <mspace width="4pt" /> <mtext>odd</mtext> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </Equation>hold for all <InlineEquation ID="IEq32"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_69_Article_IEq32.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="87" /> </InlineMediaObject> <EquationSource Format="TEX">$x,y\in (0,\pi)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>π</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq33"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_69_Article_IEq33.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="75" /> </InlineMediaObject> <EquationSource Format="TEX">$x+y\leq \pi$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>≤</mo> <mi>π</mi> </mrow> </math></EquationSource> </InlineEquation>.</p>

Inequalities for $1/(1-\cos(x) )$ and its derivatives

期刊论文
发表日期：

2025年2月24日

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H. Alzer,
H. L. Pedersen

摘要

We prove that the function $g(x)= 1 / ( 1 - \cos(x) )$ $g (x) = 1 / (1 - cos (x))$ is completely monotonic on $(0,\pi]$ $(0, π]$ and absolutely monotonic on $[\pi, 2\pi)$ $[π, 2 π)$ , and we determine the best possible bounds $\lambda_n$ $λ_{n}$ and $\mu_n$ $μ_{n}$ such that the inequalities $\lambda_n \leq g^{(n)}(x)+g^{(n)}(y)-g^{(n)}(x+y) \quad (n \geq 0 \ \mbox{even})$ $λ_{n} \leq g^{(n)} (x) + g^{(n)} (y) - g^{(n)} (x + y) (n \geq 0 even)$ and $\mu_n \leq g^{(n)}(x+y)-g^{(n)}(x)-g^{(n)}(y) \quad (n \geq 1 \ \mbox{odd})$ $μ_{n} \leq g^{(n)} (x + y) - g^{(n)} (x) - g^{(n)} (y) (n \geq 1 odd)$ hold for all $x,y\in (0,\pi)$ $x, y \in (0, π)$ with $x+y\leq \pi$ $x + y \leq π$ .

Inequalities for \(1/(1-\cos(x) )\) and its derivatives

摘要