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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Autor principal:	Ling Zhu
Formato:	article
Lenguaje:	EN
Publicado:	MDPI AG 2021
Materias:	bounds modified Bessel function of the first kind hyperbolic function mean inequality Mathematics QA1-939
Acceso en línea:	https://doaj.org/article/3efc24e84f4c47078f4312ec89f09886
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spelling	oai:doaj.org-article:3efc24e84f4c47078f4312ec89f098862021-11-25T18:16:43ZNew Bounds for the Modified Bessel Function of the First Kind and Toader-Qi Mean10.3390/math92228672227-7390https://doaj.org/article/3efc24e84f4c47078f4312ec89f098862021-11-01T00:00:00Zhttps://www.mdpi.com/2227-7390/9/22/2867https://doaj.org/toc/2227-7390Let <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.Ling ZhuMDPI AGarticleboundsmodified Bessel function of the first kindhyperbolic functionmeaninequalityMathematicsQA1-939ENMathematics, Vol 9, Iss 2867, p 2867 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	bounds modified Bessel function of the first kind hyperbolic function mean inequality Mathematics QA1-939
spellingShingle	bounds modified Bessel function of the first kind hyperbolic function mean inequality Mathematics QA1-939 Ling Zhu New Bounds for the Modified Bessel Function of the First Kind and Toader-Qi Mean
description	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.
format	article
author	Ling Zhu
author_facet	Ling Zhu
author_sort	Ling Zhu
title	New Bounds for the Modified Bessel Function of the First Kind and Toader-Qi Mean
title_short	New Bounds for the Modified Bessel Function of the First Kind and Toader-Qi Mean
title_full	New Bounds for the Modified Bessel Function of the First Kind and Toader-Qi Mean
title_fullStr	New Bounds for the Modified Bessel Function of the First Kind and Toader-Qi Mean
title_full_unstemmed	New Bounds for the Modified Bessel Function of the First Kind and Toader-Qi Mean
title_sort	new bounds for the modified bessel function of the first kind and toader-qi mean
publisher	MDPI AG
publishDate	2021
url	https://doaj.org/article/3efc24e84f4c47078f4312ec89f09886
work_keys_str_mv	AT lingzhu newboundsforthemodifiedbesselfunctionofthefirstkindandtoaderqimean
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New Bounds for the Modified Bessel Function of the First Kind and Toader-Qi Mean

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