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Bounds on differences of adjacent zeros of Bessel functions and iterative relations between consecutive zeros
 Math. Comp
"... Abstract. Bounds for the distance cν,s − c ν±1,s ′  between adjacent zeros of cylinder functions are given; s and s ′ are such that ∄c ν,s ′ ′ ∈]cν,s,c ν±1,s ′ [; cν,k stands for the kth positive zero of the cylinder (Bessel) function Cν(x) = cos αJν(x) − sin αYν(x), α ∈ [0,π[, ν ∈ R. These bou ..."
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Abstract. Bounds for the distance cν,s − c ν±1,s ′  between adjacent zeros of cylinder functions are given; s and s ′ are such that ∄c ν,s ′ ′ ∈]cν,s,c ν±1,s ′ [; cν,k stands for the kth positive zero of the cylinder (Bessel) function Cν(x) = cos αJν(x) − sin αYν(x), α ∈ [0,π[, ν ∈ R. These bounds, together with the application of modified (global) Newton methods based on the monotonic functions fν(x) =x 2ν−1 Cν(x)/Cν−1(x) and gν(x) =−x −(2ν+1) Cν(x)/Cν+1(x), give rise to forward (cν,k → cν,k+1) and backward (cν,k+1 → cν,k) iterative relations between consecutive zeros of cylinder functions. The problem of finding all the positive real zeros of Bessel functions Cν(x) for any real α and ν inside an interval [x1,x2], x1> 0, is solved in a simple way.
ON TURÁN TYPE INEQUALITIES FOR MODIFIED BESSEL FUNCTIONS
"... Abstract. In this note our aim is to point out that certain inequalities for modified Bessel functions of the first and second kind, deduced recently by Laforgia and Natalini, are in fact equivalent to the corresponding Turán type inequalities for these functions. Moreover, we present some new Turán ..."
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Abstract. In this note our aim is to point out that certain inequalities for modified Bessel functions of the first and second kind, deduced recently by Laforgia and Natalini, are in fact equivalent to the corresponding Turán type inequalities for these functions. Moreover, we present some new Turán type inequalities for the aforementioned functions and we show that their product is decreasing as a function of the order, which has an application in the study of stability of radially symmetric solutions in a generalized FitzHughNagumo equation in two spatial dimensions. At the end of this note an open problem is posed, which may be of interest for further research. 1. Some inequalities for modified Bessel functions Let us denote with Iν and Kν the modified Bessel functions of the first and second kind, respectively. For definitions, recurrence formulas and other properties of modified Bessel functions of the first and second kind we refer to the classical book of Watson [35]. In 2007, motivated by a problem which arises in biophysics, Penfold et al. [31]