Some monotonicity results for ratios of modified Bessel functions
Authors:
Henry C. Simpson and Scott J. Spector
Journal:
Quart. Appl. Math. 42 (1984), 95-98
MSC:
Primary 33A40; Secondary 73G05
DOI:
https://doi.org/10.1090/qam/736509
MathSciNet review:
736509
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Abstract: We consider the functions ${v_\alpha }(t) \equiv t{I_\alpha }(t)/{I_{\alpha + 1}}(t)$ where ${I_\alpha }$ are the modified Bessel functions of the first kind of order $\alpha \ge 0$. We prove that ${v_\alpha }$ is strictly monotone and strictly convex on R$^{+}$ . These results have application in finite elasticity$^{1}$.
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S. Y. Cheng, S. T. Ariaratnam and R. N. Dubey, Axisymmetric bifurcation in an elastic-plastic cylinder under axial load and lateral hydrostatic pressure, Quart. Appl. Math. 29, 41–51 (1971)
- James Alan Cochran, The monotonicity of modified Bessel functions with respect to their order, J. Math. and Phys. 46 (1967), 220–222. MR 213624
- A. E. Green and A. J. M. Spencer, The stability of a circular cylinder under finite extension and torsion, Math. Phys. 37 (1959), 316–338. MR 0101660
- Mourad E. H. Ismail and Douglas H. Kelker, Special functions, Stieltjes transforms and infinite divisibility, SIAM J. Math. Anal. 10 (1979), no. 5, 884–901. MR 541088, DOI https://doi.org/10.1137/0510083
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J. P. Miles, Bifurcation in plastic flow under uniaxial tension, J. Mech. Phys. Solids 19, 89–102 (1971)
- Ingemar Nȧsell, Inequalities for modified Bessel functions, Math. Comp. 28 (1974), 253–256. MR 333288, DOI https://doi.org/10.1090/S0025-5718-1974-0333288-9
- Chester B. Sensenig, Instability of thick elastic solids, Comm. Pure Appl. Math. 17 (1964), 451–491. MR 169453, DOI https://doi.org/10.1002/cpa.3160170406
- Henry C. Simpson and Scott J. Spector, On barrelling for a special material in finite elasticity, Quart. Appl. Math. 42 (1984), no. 1, 99–111. MR 736510, DOI https://doi.org/10.1090/S0033-569X-1984-0736510-2
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- G. N. Watson, A treatise on the theory of Bessel functions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995. Reprint of the second (1944) edition. MR 1349110
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D. E. Amos, Computation of modified Bessel functions and their ratios, Math. Comp. 28, 239–251 (1974)
S. Y. Cheng, S. T. Ariaratnam and R. N. Dubey, Axisymmetric bifurcation in an elastic-plastic cylinder under axial load and lateral hydrostatic pressure, Quart. Appl. Math. 29, 41–51 (1971)
J. A. Cochran, The monotonicity of modified Bessel functions with respect to their order, J. Math. and Phys. 46, 220–222 (1967)
A. E. Green and A. J. M. Spencer, The stability of a circular cylinder under finite extension and torsion, J. Math. and Phys. 37, 316–338 (1959)
M. E. H. Ismail and D. H. Kelker, Special functions, Stieltjes transforms and infinite divisibility, SIAM J. Math. Anal. 10, 884–901 (1979)
A. L. Jones, An extension of an inequality involving modified Bessel functions, J. Math. and Phys. 47, 220–221 (1968)
J. P. Miles, Bifurcation in plastic flow under uniaxial tension, J. Mech. Phys. Solids 19, 89–102 (1971)
I. Nasell, Inequalities for modified Bessel functions, Math. Comp. 28, 253–256 (1974)
C. B. Sensenig, Instability of thick elastic solids. Comm. Pure Appl. Math. 17, 451–491 (1964)
H. C. Simpson and S. J. Spector, On barrelling for a special material in finite elasticity. Quart. Appl. Math. This issue.
R. P. Soni, On an inequality for modified Bessel functions, J. Math. and Phys. 44, 406–407 (1965)
G. N. Watson, A treatise on the theory of Bessel functions, Cambridge Univ. Press (1922)
E. W. Wilkes, On the stability of a circular tube under end thrust, Quart. J. Mech. Appl. Math. 8, 88–100 (1955)
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Article copyright:
© Copyright 1984
American Mathematical Society