Inequalities and monotonicity results for zeros of modified Bessel functions of purely imaginary order

Author:
Andrea Laforgia

Journal:
Quart. Appl. Math. **44** (1986), 91-96

MSC:
Primary 33A40; Secondary 34C10

DOI:
https://doi.org/10.1090/qam/840446

MathSciNet review:
840446

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Abstract: Let and denote the th positive zeros, in decreasing order, of the modified Bessel function of purely imaginary order and of its derivative , respectively. We show that for and and and decrease as increases. Some related results are mentioned for the zeros of and the chain of inequalities is established.

**[1]**M. Abramowitz and I. A. Stegun, eds.,*Handbook of mathematical functions*, Applied Mathematics Series**55**, National Bureau of Standards, Washington, 1964**[2]**S. Ahmed, A. Laforgia, and M. E. Muldoon,*On the spacing of the zeros of some classical orthogonal polynomials*, J. London Math. Soc. (2)**25**(1982), no. 2, 246–252. MR**653383**, https://doi.org/10.1112/jlms/s2-25.2.246**[3]**Á. Elbert and A. Laforgia,*Some monotonicity properties of the zeros of ultraspherical polynomials*, Acta Math. Hungar.**48**(1986), no. 1-2, 155–159. MR**858393**, https://doi.org/10.1007/BF01949060**[4]**Erasmo M. Ferreira and Javier Sesma,*Zeros of the modified Hankel function*, Numer. Math.**16**(1970), 278–284. MR**0271418**, https://doi.org/10.1007/BF02219779**[5]**A. Gray, G. B. Mathews and T. M. Macrobert,*A Treatise on Bessel Functions and their Applications to Physics*, Macmillan, London, 1952**[6]**E. L. Ince,*Ordinary Differential Equations*, Dover Publications, New York, 1944. MR**0010757****[7]**Andrea Laforgia,*Sturm theory for certain classes of Sturm-Liouville equations and Turánians and Wronskians for the zeros of derivative of Bessel functions*, Nederl. Akad. Wetensch. Indag. Math.**44**(1982), no. 3, 295–301. MR**674902****[8]**J. T. Lewis and M. E. Muldoon,*Monotonicity and convexity properties of zeros of Bessel functions*, SIAM J. Math. Anal.**8**(1977), no. 1, 171–178. MR**0437823**, https://doi.org/10.1137/0508012**[9]**Lee Lorch,*Elementary comparison techniques for certain classes of Sturm-Liouville equations*, Differential equations (Proc. Internat. Conf., Uppsala, 1977) Almqvist & Wiksell, Stockholm, 1977, pp. 125–133. Sympos. Univ. Upsaliensis Ann. Quingentesimum Celebrantis, No. 7. MR**0492536****[10]**G. Szegö,*Orthogonal Polynomials*, Amer. Math. Soc. Colloq. Publ.**23**, 4th ed., Amer. Math. Soc., Providence, R. I., 1975**[11]**G. Szegö and P. Turán,*On the monotone convergence of certain Riemann sums*, Publ. Math. Debrecen**8**(1961), 326–335. MR**0137818****[12]**G. N. Watson,*A Treatise on the Theory of Bessel Functions*, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. MR**0010746**

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Additional Information

DOI:
https://doi.org/10.1090/qam/840446

Article copyright:
© Copyright 1986
American Mathematical Society