New inequalities from classical Sturm theorems |
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Authors: | Alfredo Deao Amparo Gil Javier Segura |
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Affiliation: | aDepartamento de Matemáticas, Universidad Carlos III de Madrid, 28911 Leganés (Madrid), Spain;bDepartamento de Matemáticas, Estadística y Computación, Facultad de Ciencias, Universidad de Cantabria, Avda. de Los Castros, s/n, 39005 Santander, Spain |
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Abstract: | Inequalities satisfied by the zeros of the solutions of second-order hypergeometric equations are derived through a systematic use of Liouville transformations together with the application of classical Sturm theorems. This systematic study allows us to improve previously known inequalities and to extend their range of validity as well as to discover inequalities which appear to be new. Among other properties obtained, Szegő's bounds on the zeros of Jacobi polynomials for , are completed with results for the rest of parameter values, Grosjean's inequality (J. Approx. Theory 50 (1987) 84) on the zeros of Legendre polynomials is shown to be valid for Jacobi polynomials with |β|1, bounds on ratios of consecutive zeros of Gauss and confluent hypergeometric functions are derived as well as an inequality involving the geometric mean of zeros of Bessel functions. |
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Keywords: | Sturm comparison theorem Hypergeometric functions Orthogonal polynomials |
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