Bounds for Turánians of modified Bessel functions

Abstract

Abstract. 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 kind are deduced. These inequalities provide sharp lower and upper bounds for the Turánian of modified Bessel functions of the first and second kind, 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 Gronwall [20] on ordinary differential equations, an integral representation of Ismail [29,30] for the quotient of modified Bessel functions of the second kind and some results of Hartman and Watson [25,27,62]. 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 kind and it is shown that this product is strictly geometrically concave.

Bounds for modified Lommel functions of the first kind and their ratios

“…Table 3: Relative error in approximatingt µ,ν (x)/t µ−1,ν−1 (x) by the lower bound of (3.38). 0.0101 0.0313 0.0593 0.0285 0.0122 0.0067 0.0031 0.0011 0.0003 (2,2.5) 0.0015 0.0054 0.0200 0.0244 0.0144 0.0093 0.0049 0.0020 0.0006 (4.5,5) 0.0003 0.0010 0.0050 0.0105 0.0106 0.0088 0.0058 0.0029 0.0009 (9.5,10) 0.0000 0.0002 0.0009 0.0027 0.0041 0.0047 0.0046 0.0033 0.0014 (2,0) 0.0495 0.1201 0.1935 0.1197 0.0360 0.0086 0.0013 0.0004 0.0001 (3,1) 0.0090 0.0308 0.0941 0.0932 0.0440 0.0156 0.0034 0.0011 0.0003 (4.5,2.5) 0.0020 0.0077 0.0343 0.0559 0.0407 0.0206 0.0056 0.0020 0.0006 (7,5) 0.0005 0.0018 0.0097 0.0238 0.0263 0.0196 0.0075 0.0029 0.0009 (12,10) 0.0000 0.0003 0.0019 0.0062 0.0097 0.0107 0.0073 0.0034 0.0014 (5,0) 0.0284 0.0756 0.1715 0.1936 0.1302 0.0586 0.0054 0.0004 0.0001 (6,1) 0.0070 0.0247 0.0892 0.1349 0.1116 0.0635 0.0097 0.0012 0.0003 (7.5,2.5) 0.0020 0.0077 0.0371 0.0778 0.0816 0.0595 0.0151 0.0020 0.0006 (10,5) 0.0005 0.0022 0.0120 0.0336 0.0458 0.0436 0.0195 0.0031 0.0009 (15,10) 0.0001 0.0005 0.0027 0.0092 0.0160 0.0200 0.0169 0.0041 0.0014…”

Sharp Bounds for the Ratio of Modified Bessel Functions

“…Other results concerning Amos type inequality or Simpson-Spector type inequality can be found in [25], [5], [6], [7], [8] and references therein..…”

On the Capacity of the Peak Power Constrained Vector Gaussian Channel: An Estimation Theoretic Perspective

“…To show the second part of Theorem 2 let R = c √ n. We will also need the following bounds on h v (x) [18], [19]:…”