<p>In this paper, we consider the function <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$I_{\nu }(x)\textbf{L}_{\nu -1}(x) - I_{\nu -1}(x)\textbf{L}_{\nu }(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>I</mi> <mi>ν</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mi mathvariant="bold">L</mi> <mrow> <mi>ν</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <msub> <mi>I</mi> <mrow> <mi>ν</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mi mathvariant="bold">L</mi> <mi>ν</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$I_\nu (x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>I</mi> <mi>ν</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\textbf{L}_\nu (x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="bold">L</mi> <mi>ν</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> denote the modified Bessel and Struve functions of the first kind, respectively. We first derive a series representation for this function and then use it to establish sharp bounds for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$I_{\nu }(x)\textbf{L}_{\nu -1}(x) - I_{\nu -1}(x)\textbf{L}_{\nu }(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>I</mi> <mi>ν</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mi mathvariant="bold">L</mi> <mrow> <mi>ν</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <msub> <mi>I</mi> <mrow> <mi>ν</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mi mathvariant="bold">L</mi> <mi>ν</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, improving the results given by Gaunt (J Math Anal Appl 468(1):547–566, 2018). Finally, we obtain some new bounds for the difference <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\textbf{L}_{\nu -1}(x)/\textbf{L}_{\nu }(x) - I_{\nu -1}(x)/I_{\nu }(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="bold">L</mi> <mrow> <mi>ν</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">/</mo> <msub> <mi mathvariant="bold">L</mi> <mi>ν</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <msub> <mi>I</mi> <mrow> <mi>ν</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">/</mo> <msub> <mi>I</mi> <mi>ν</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, which allow us to obtain refined bounds for <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\textbf{L}_{\nu -1}(x)/\textbf{L}_{\nu }(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="bold">L</mi> <mrow> <mi>ν</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">/</mo> <msub> <mi mathvariant="bold">L</mi> <mi>ν</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with the help of existing sharp bounds for <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$I_{\nu -1}(x)/I_{\nu }(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>I</mi> <mrow> <mi>ν</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">/</mo> <msub> <mi>I</mi> <mi>ν</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.</p>

Sharp Bounds for the Ratio of Modified Struve Functions of the First Kind

期刊论文
发表日期：

2025年11月5日

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Zhong-Xuan Mao,
Jing-Feng Tian

摘要

In this paper, we consider the function $I_{\nu }(x)\textbf{L}_{\nu -1}(x) - I_{\nu -1}(x)\textbf{L}_{\nu }(x)$ $I_{ν} (x) L_{ν - 1} (x) - I_{ν - 1} (x) L_{ν} (x)$ , where $I_\nu (x)$ $I_{ν} (x)$ and $\textbf{L}_\nu (x)$ $L_{ν} (x)$ denote the modified Bessel and Struve functions of the first kind, respectively. We first derive a series representation for this function and then use it to establish sharp bounds for $I_{\nu }(x)\textbf{L}_{\nu -1}(x) - I_{\nu -1}(x)\textbf{L}_{\nu }(x)$ $I_{ν} (x) L_{ν - 1} (x) - I_{ν - 1} (x) L_{ν} (x)$ , improving the results given by Gaunt (J Math Anal Appl 468(1):547–566, 2018). Finally, we obtain some new bounds for the difference $\textbf{L}_{\nu -1}(x)/\textbf{L}_{\nu }(x) - I_{\nu -1}(x)/I_{\nu }(x)$ $L_{ν - 1} (x) / L_{ν} (x) - I_{ν - 1} (x) / I_{ν} (x)$ , which allow us to obtain refined bounds for $\textbf{L}_{\nu -1}(x)/\textbf{L}_{\nu }(x)$ $L_{ν - 1} (x) / L_{ν} (x)$ with the help of existing sharp bounds for $I_{\nu -1}(x)/I_{\nu }(x)$ $I_{ν - 1} (x) / I_{ν} (x)$ .