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Bounds for ratios of modified Bessel functions and associated Turán-type inequalities

Abstract: ABSTRACT: New sharp inequalities for the ratios of Bessel functions of consecutive orders are obtained using as main tool the first order difference-differential equations satisfied by these functions; many already known inequalities are also obtainable, and most of them can be either improved or the range of validity extended. It is shown how to generate iteratively upper and lower bounds, which are converging sequences in the case of the I-functions. Few iterations provide simple and effective upper and lower bounds for approximating The ratios I? (x)/I??1(x) and the condition numbers xI ? (x)/I? (x) for any x, ? 0; for the ratios K? (x)/K?+1(x) the same is possible, but with some restrictions on ?. Using these bounds Turán-type inequalities are established, extending the range of validity of some known inequalities and obtaining new inequalities as well; for instance, it is shown that K?+1(x)K??1(x)/(K? (x))2 < |?|/(|?| ? 1), x > 0, ? ? [? / 1, 1] and that the inequality is the best possible; this proves and improves an existing conjecture.

 Autoría: Segura J.,

 Fuente: Journal of Mathematical Analysis and Applications, 2011, 374(2), 516-528

 Editorial: Academic Press Inc.

 Fecha de publicación: 15/02/2011

 Nº de páginas: 13

 Tipo de publicación: Artículo de Revista

 DOI: 10.1016/j.jmaa.2010.09.030

 ISSN: 0022-247X,1096-0813

 Proyecto español: MTM2009-11686

 Url de la publicación: https://doi.org/10.1016/j.jmaa.2010.09.030