Positive Trigonometric Integrals Associated with Some Lommel Functions of the First Kind

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• 作者：Stamatis Koumandos
• 关键词：Lommel functions ; trigonometric integrals ; monotonicity properties ; inequalities ; zeros
• 刊名：Mediterranean Journal of Mathematics
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：14
• 期：1
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics, general;
• 出版者：Springer International Publishing
• ISSN：1660-5454
• 卷排序：14

文摘

We prove that $$\begin{aligned} \int _{0}^{x} t^{\mu -\frac{1}{2}}\,\log t\, \sin (x-t)\,\mathrm{d}t=\sqrt{x}\,\frac{\mathrm{d}}{\mathrm{d}\mu }\,s_{\mu ,\frac{1}{2}}(x)>0, \end{aligned}$$for all \(\mu \ge 1/2\) and \(x\ge \pi \), where \(s_{\mu ,\frac{1}{2}}(x)\) is the Lommel function of the first kind. This refines a result of Steinig published in 1972. As an application, we derive positive functional bounds for the functions \(s_{\mu ,\frac{1}{2}}(x)\), \(\mu \ge 1/2\) and \(x\ge \pi \). Furthermore, we give estimates for the location of the zeros of the functions \(s_{\mu ,1/2}(x)\), \(-3/2<\mu <1/2\), \(\mu \ne -1/2\), \(x>0\).KeywordsLommel functionstrigonometric integralsmonotonicity propertiesinequalitieszerosMathematics Subject ClassificationPrimary 33C1033B10Secondary 26D1533B15References1.Andrews, G.E., Askey, R., Roy, R.: Special Functions. Cambridge Univ. 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Princeton University Press, Princeton (1946)MATHGoogle ScholarCopyright information© Springer International Publishing 2016Authors and AffiliationsStamatis Koumandos1Email author1.Department of Mathematics and StatisticsThe University of CyprusNicosiaCyprus About this article CrossMark Publisher Name Springer International Publishing Print ISSN 1660-5446 Online ISSN 1660-5454 About this journal Reprints and Permissions Article actions .buybox { margin: 16px 0 0; position: relative; } .buybox { font-family: Source Sans Pro, Helvetica, Arial, sans-serif; font-size: 14px; font-size: .875rem; } .buybox { zoom: 1; } .buybox:after, .buybox:before { content: ''; display: table; } .buybox:after { clear: both; } /*---------------------------------*/ .buybox .buybox__header { border: 1px solid #b3b3b3; border-bottom: 0; padding: 8px 12px; position: relative; background-color: #f2f2f2; } .buybox__header .buybox__login { font-family: Source Sans Pro, Helvetica, Arial, sans-serif; font-size: 14px; 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