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Some inequalities of Bessel and modified Bessel functions

Part of: Inequalities

Published online by Cambridge University Press:  09 April 2009

C. M. Joshi  and
S. K. Bissu
C. M. Joshi
Affiliation:
Sukhadia UniversityUdaipur 313001, India
S. K. Bissu
Affiliation:
Sukhadia UniversityUdaipur 313001, India
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Abstract

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Two-sided inequalties for the ratio of modified Bessel functions of first kind are given, which provide sharper upper and lower bounds than had been known earlier. Wronskian type inequalities for Bessel functions are proved, and in the sequel alternative proofs of Turan-type inequalities for Bessel and modified Bessel functions are also discussed. These then lead to a two-sided inequality for Bessel functions. Also incorporated in the discussion is an inequality for the ratio of two Bessel functions for 0 < x < 1. Verifications of these inequalities are pointed out numerically.

Keywords

Bessel and modified Bessel functionsTuran and Wronski type inequalities

MSC classification

Secondary: 26D99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 50 , Issue 2 , April 1991 , pp. 333 - 342
Copyright
Copyright © Australian Mathematical Society 1991

References

[1]Cochran, J. A., ‘The monotonicity of modified Bessel functions with respect to their order’, J. Math. Phys. 46 (1967), 220–222.CrossRefGoogle Scholar
[2]Erdélyi, A., Higher transcendental functions, Vol. II, (McGraw-Hill, New York, 1953).Google Scholar
[3]Jones, A. L., ‘An extension of an inequality involving modified Bessel functions’, J. Math. Phys. 47 (1968), 220–221.CrossRefGoogle Scholar
[4]Luke, Y. L., Integrals of Bessel functions, (McGraw-Hill, New York, 1962).Google Scholar
[5]Nάsell, I., ‘Inequalities for modified Bessel functions’, Math. Comp. 28 (1975), 253–256.Google Scholar
[6]Rosenthal, D. K., ‘The shape and stability of a bubble at the axis of a rotating liquid’, J. Fluid Mech. 12 (1962), 358–366.CrossRefGoogle Scholar
[7]Ross, D. K., ‘The stability of a column of liquid to torsial oscillations’, Z. Angew. Math. Phys. 21 (1970), 137–140.CrossRefGoogle Scholar
[8]Ross, D. K., ‘Inequalities for special functions’, Problem 72-15, SIAM Rev. 15 (1973), 668–670.Google Scholar
[9]Skovgaard, H., ‘On inequalities of Turan type’, Math. Scand. 2 (1954), 65–73.Google Scholar
[10]Soni, R. P., ‘On an inequality for modified Bessel functions’, J. Math. Phys. 44 (1965), 406–407.CrossRefGoogle Scholar
[11]Szàsz, O., ‘Inequalities concerning ultraspherical polynomials and Bessel functions’, Proc. Amer. Math. Soc. 1 (1950), 256–267.CrossRefGoogle Scholar
[12]Thiruvenkatachar, V. K., and Najundiah, T. S., ‘Inequalities concerning Bessel functions and orthogonal polynomials’, Proc. Indian Nat. Acad. Part A 33 (1951), 373–384.Google Scholar
[13]Watson, G. N., A treatise on the theory of Bessel functions, 2nd ed., (Cambridge Univ. Press, 1944).Google Scholar