Unimodality preservation by ratios of functional series and integral transforms
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| Format: | Preprint |
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2024
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| _version_ | 1866913713876893696 |
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| author | Karp, Dmitrii Vishnyakova, Anna Zhang, Yi |
| author_facet | Karp, Dmitrii Vishnyakova, Anna Zhang, Yi |
| contents | An elementary, but very useful lemma due to Biernacki and Krzyż (1955) asserts that the ratio of two power series inherits monotonicity from that of the sequence of ratios of their respective coefficients. Over the last two decades it has been realized that, under some additional assumptions, similar claims hold for more general series ratios as well as for unimodality in place of monotonicity. This paper continues this line of research: we consider ratios of general functional series and integral transforms and furnish natural sufficiency conditions for preservation of unimodality by such ratios. Numerous series and integral transforms appearing in applications satisfy our sufficiency conditions, including Dirichlet, factorial and inverse factorial series, Laplace, Mellin and generalized Stieltjes transforms, among many others. Finally, we illustrate our general results by exhibiting certain statements on monotonicity patterns for ratios of some special functions. The key role in our considerations is played by the notion of sign regularity. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_01755 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Unimodality preservation by ratios of functional series and integral transforms Karp, Dmitrii Vishnyakova, Anna Zhang, Yi Classical Analysis and ODEs Primary 26A48, 44A05, Secondary 15B48, 33C20, 33D15, 33E20 An elementary, but very useful lemma due to Biernacki and Krzyż (1955) asserts that the ratio of two power series inherits monotonicity from that of the sequence of ratios of their respective coefficients. Over the last two decades it has been realized that, under some additional assumptions, similar claims hold for more general series ratios as well as for unimodality in place of monotonicity. This paper continues this line of research: we consider ratios of general functional series and integral transforms and furnish natural sufficiency conditions for preservation of unimodality by such ratios. Numerous series and integral transforms appearing in applications satisfy our sufficiency conditions, including Dirichlet, factorial and inverse factorial series, Laplace, Mellin and generalized Stieltjes transforms, among many others. Finally, we illustrate our general results by exhibiting certain statements on monotonicity patterns for ratios of some special functions. The key role in our considerations is played by the notion of sign regularity. |
| title | Unimodality preservation by ratios of functional series and integral transforms |
| topic | Classical Analysis and ODEs Primary 26A48, 44A05, Secondary 15B48, 33C20, 33D15, 33E20 |
| url | https://arxiv.org/abs/2408.01755 |