Bounds for Turánians of modified Bessel functions. (English) Zbl 1316.33002
Summary: Motivated by some applications in applied mathematics, biology, chemistry, physics and engineering sciences, new tight Turán type inequalities for modified Bessel functions of the first and second kinds are deduced. These inequalities provide sharp lower and upper bounds for the Turánian of modified Bessel functions of the first and second kinds, and in most cases the relative errors of the bounds tend to zero as the argument tends to infinity. The chief tools in our proofs are some ideas of T. H. Gronwall [Ann. Math. (2) 33, 275–278 (1932; Zbl 0004.21101)] on ordinary differential equations, an integral representation of M. E. H. Ismail [Ann. Probab. 5, 582–585 (1977; Zbl 0369.60023); Proc. Am. Math. Soc. 108, No. 2, 353–361 (1990; Zbl 0685.33004)] for the quotient of modified Bessel functions of the second kind and some results of Hartman and Watson [P. Hartman, Am. J. Math. 83, 154–188 (1961; Zbl 0096.27001); P. Hartman and G. S. Watson, Ann. Probab. 2, 593–607 (1974; Zbl 0305.60033); G. S. Watson, Statistics on spheres. The University of Arkansas Lecture Notes in the Mathematical Sciences, Vol. 6. A Wiley-Interscience Publication. New York etc.: John Wiley & Sons (1983; Zbl 0646.62045)]. As applications of the main results some sharp Turán type inequalities are presented for the product of modified Bessel functions of the first and second kinds and it is shown that this product is strictly geometrically concave.
MSC:
| 33C10 | Bessel and Airy functions, cylinder functions, \({}_0F_1\) |
| 39B62 | Functional inequalities, including subadditivity, convexity, etc. |
Keywords:
modified Bessel functions of the first and second kinds; Bessel functions of the first and second kinds; product of modified Bessel functions; sharp bounds; Turán-type inequalities; functional inequalitiesCitations:
Zbl 0004.21101; Zbl 0369.60023; Zbl 0685.33004; Zbl 0096.27001; Zbl 0305.60033; Zbl 0646.62045Software:
DLMFReferences:
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