Inequalities involving Bessel and modified Bessel functions

Inequalities for integrals of modified Bessel functions and expressions involving them

“…The We have equality in (A.56) if and only if ν = 1 2 . The inequalities for K ν (x) can be found in [22], whilst the inequality for I ν (x) can be found in [23] and [27], which extends a result of [34]. A survey of related inequalities for modified Bessel functions is given by [7], and lower and upper bounds for the ratios Iν (x) I ν−1 (x) and Kν (x) K ν−1 (x) , which improve on inequalities (A.54) -(A.56), are also given in [22] and [33].…”

Inequalities for the modified Bessel function of the second kind and the kernel of the Krätzel integral transformation

“…Proof. The upper bound holds because K ν (x) < K 1/2 (x) = π 2x e −x for all ν < 1 2 (see [5]). The lower bound follows from Theorem 2.1 and an application of the inequality Γ(x+a) Γ(x+1) > 1 (x+a) 1−a for 0 < a < 1 (see [4]).…”

A Re-entrant Phase Transition in the Survival of Secondary Infections on Networks

“…Ismail and Muldoon [ 15 ] list many different bounds on , including those in ( 18 ) coming from formulas (6.7) and (6.22) in that article. For the second part, it is well-known [ 1 ] that the Bessel functions of the first kind may be expressed as hence, introducing , we obtain This function has previously been studied by Ifantis and Siafarikas [ 14 ], who proved various inequalities including their formulas (1.2) and (2.17) which imply the lower and upper bounds of ( 19 ). …”