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Báo cáo hóa học: " Research Article Extension of Oppenheim’s Problem to Bessel Functions"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Extension of Oppenheim’s Problem to Bessel Functions | Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2007 Article ID 82038 7 pages doi 10.1155 2007 82038 Research Article Extension of Oppenheim s Problem to Bessel Functions Arpad Baricz and Ling Zhu Received 10 September 2007 Accepted 22 October 2007 Recommended by Andrea Laforgia Our aim is to extend some trigonometric inequalities to Bessel functions. Moreover we deduce the hyperbolic analogue of these trigonometric inequalities and we extend these inequalities to modified Bessel functions. Copyright 2007 A. Baricz and L. Zhu. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction and main results In 1957 Ogilvy et al. 1 or see 2 page 238 asked the following question for each a1 0 there is a greatest a2 and a least a3 such that for all x e 0 n 2 the inequality sin x sin x . a2 --------- x a3 --------- 1.1 1 a1cos x 1 a1cos x holds. Determine a2 and a3 as functions of a1. In 1958 Oppenheim and Carver 3 or see 2 page 238 gave a partial solution to Oppenheim s problem by showing that for all a1 e 0 1 2 and x e 0 n 2 1.1 holds when a2 a1 1 and a3 n 2. Recently Zhu 4 Theorem 7 solved completely this problem of Oppenheim proving that 1.1 holds in the following cases i if a1 e 0 1 2 then a2 a1 1 and a3 n 2 ii if a1 e 1 2 n 2 - 1 then a2 4a1 1 - a2 and a3 n 2 iii if a1 e n 2 - 1 2 n then a2 4a1 1 - a2 and a3 a1 1 iv if a1 2 n then a2 n 2 and a3 a1 1 where a2 and a3 are the best constants in i and iv while a3 is also the best constant in ii and iii . Recently Baricz 5 Theorem 2.20 extended the Carver solution to Bessel functions see also 6 for further results . In this note our aim is to extend the above-mentioned 2 Journal of Inequalities and Applications Zhu solution to Bessel functions too. For this let us consider the function fp R -ro 1 defined by fp x 2pr p 1 x-pJp x X