Let <InlineEquation ID="IEq1"><EquationSource Format="MATHML"><math><mi>R</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></math></EquationSource><EquationSource Format="TEX">$R\left ( x\right ) $</EquationSource></InlineEquation> be the Mills ratio of the standard Gaussian law. This paper investigates the higher order monotonicity of the function <Equation ID="Equa"><EquationSource Format="MATHML"><math><mi>x</mi><mo>↦</mo><msub><mi>Y</mi><mrow><mi>a</mi><mo>,</mo><mi>b</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>a</mi><mo>)</mo></mrow><msup><mi>R</mi><mn>2</mn></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>+</mo><mi>b</mi><mi>x</mi><mi>R</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></math></EquationSource><EquationSource Format="TEX">$ x\mapsto Y_{a,b}\left ( x\right ) =\left ( x^{2}+a\right ) R^{2} \left ( x\right ) +bxR\left ( x\right ) $</EquationSource></Equation>on <InlineEquation ID="IEq2"><EquationSource Format="MATHML"><math><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi mathvariant="normal">∞</mi><mo>)</mo></mrow></math></EquationSource><EquationSource Format="TEX">$\left ( 0,\infty \right ) $</EquationSource></InlineEquation>, where <InlineEquation ID="IEq3"><EquationSource Format="MATHML"><math><mi>a</mi><mo>,</mo><mi>b</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi></math></EquationSource><EquationSource Format="TEX">$a,b\in \mathbb{R}$</EquationSource></InlineEquation>. In particular, we prove that the function <InlineEquation ID="IEq4"><EquationSource Format="MATHML"><math><mi>x</mi><mo>↦</mo><msub><mi>Y</mi><mrow><mi>a</mi><mo>,</mo><mi>a</mi><mo>−</mo><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>−</mo><mi>a</mi><mo>+</mo><mn>1</mn></math></EquationSource><EquationSource Format="TEX">$x\mapsto Y_{a,a-2}\left ( x\right ) -a+1$</EquationSource></InlineEquation> is completely monotonic on <InlineEquation ID="IEq5"><EquationSource Format="MATHML"><math><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi mathvariant="normal">∞</mi><mo>)</mo></mrow></math></EquationSource><EquationSource Format="TEX">$\left ( 0,\infty \right ) $</EquationSource></InlineEquation> if and only if <InlineEquation ID="IEq6"><EquationSource Format="MATHML"><math><mi>a</mi><mo>≥</mo><mo>−</mo><mn>1</mn></math></EquationSource><EquationSource Format="TEX">$a\geq -1$</EquationSource></InlineEquation>. As a special case, <InlineEquation ID="IEq7"><EquationSource Format="MATHML"><math><mi>x</mi><mo>↦</mo><msub><mi>Y</mi><mrow><mo>−</mo><mn>1</mn><mo>,</mo><mo>−</mo><mn>3</mn></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>+</mo><mn>2</mn></math></EquationSource><EquationSource Format="TEX">$x\mapsto Y_{-1,-3}\left ( x\right ) +2$</EquationSource></InlineEquation> is completely monotonic on <InlineEquation ID="IEq8"><EquationSource Format="MATHML"><math><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi mathvariant="normal">∞</mi><mo>)</mo></mrow></math></EquationSource><EquationSource Format="TEX">$\left ( 0,\infty \right ) $</EquationSource></InlineEquation>, which implies the complete monotonicity of the functions <InlineEquation ID="IEq9"><EquationSource Format="MATHML"><math><msup><mi>R</mi><mn>3</mn></msup><msup><mrow><mo>(</mo><mn>1</mn><mo stretchy="false">/</mo><mi>R</mi><mo>)</mo></mrow><mrow><mo>″</mo></mrow></msup></math></EquationSource><EquationSource Format="TEX">$R^{3}\left ( 1/R\right ) ^{\prime \prime }$</EquationSource></InlineEquation> and <InlineEquation ID="IEq10"><EquationSource Format="MATHML"><math><mo>−</mo><msup><mi>R</mi><mn>3</mn></msup><msup><mrow><mo>(</mo><mo>ln</mo><mi>R</mi><mo>)</mo></mrow><mrow><mo>‴</mo></mrow></msup></math></EquationSource><EquationSource Format="TEX">$-R^{3}\left ( \ln R\right ) ^{\prime \prime \prime }$</EquationSource></InlineEquation> on the same interval. As applications, several functional inequalities are derived, and some known and new sharp bounds for R are established.

Higher order monotonicity involving the Mills ratio with applications

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2026年6月19日

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Hui-Zuo Xu,
Zhen-Hang Yang

摘要

Let $R (x)$ $R\left ( x\right ) $ be the Mills ratio of the standard Gaussian law. This paper investigates the higher order monotonicity of the function $x \mapsto Y_{a, b} (x) = (x^{2} + a) R^{2} (x) + b x R (x)$ $ x\mapsto Y_{a,b}\left ( x\right ) =\left ( x^{2}+a\right ) R^{2} \left ( x\right ) +bxR\left ( x\right ) $on $(0, \infty)$ $\left ( 0,\infty \right ) $, where $a, b \in R$ $a,b\in \mathbb{R}$. In particular, we prove that the function $x \mapsto Y_{a, a - 2} (x) - a + 1$ $x\mapsto Y_{a,a-2}\left ( x\right ) -a+1$ is completely monotonic on $(0, \infty)$ $\left ( 0,\infty \right ) $ if and only if $a \geq - 1$ $a\geq -1$. As a special case, $x \mapsto Y_{- 1, - 3} (x) + 2$ $x\mapsto Y_{-1,-3}\left ( x\right ) +2$ is completely monotonic on $(0, \infty)$ $\left ( 0,\infty \right ) $, which implies the complete monotonicity of the functions $R^{3} {(1 / R)}^{″}$ $R^{3}\left ( 1/R\right ) ^{\prime \prime }$ and $- R^{3} {(ln R)}^{‴}$ $-R^{3}\left ( \ln R\right ) ^{\prime \prime \prime }$ on the same interval. As applications, several functional inequalities are derived, and some known and new sharp bounds for R are established.