New Bounds for the Modified Bessel Function of the First Kind and Toader-Qi Mean
Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>I</mi><mi>p</mi></msub><mfenced open="(" close=")"><mi>x</mi></mfe...
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| Format: | article |
| Langue: | EN |
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MDPI AG
2021
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| Accès en ligne: | https://doaj.org/article/3efc24e84f4c47078f4312ec89f09886 |
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| Résumé: | Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>I</mi><mi>p</mi></msub><mfenced open="(" close=")"><mi>x</mi></mfenced></mrow></semantics></math></inline-formula> be the modified Bessel function of the first kind of order <i>p</i>. The upper and lower bounds in the form of simple rational functions about <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo form="prefix">cosh</mo><mi>t</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mo form="prefix">sinh</mo><mi>t</mi><mo stretchy="false">)</mo><mo>/</mo><mi>t</mi></mrow></semantics></math></inline-formula> for the function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>I</mi><mn>0</mn></msub><mfenced open="(" close=")"><mi>x</mi></mfenced></mrow></semantics></math></inline-formula> are obtained. The corresponding inequalities for the Toader-Qi mean do not match those in the existing literature. |
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