Bounds for Bessel functions

We establish lower and upper bounds for the Bessel functionJ _v (x) and the modified Bessel functionI _v(x) of the first kind. Our chief tool is the differential equation satisfied by these functions.     

Bounds for ratios of modified Bessel functions and associated Turán-type inequalities

New sharp inequalities for the ratios of Bessel functions of consecutive orders are obtained using as main tool the first order difference-differential equations satisfied by these functions; many already known inequalities are also obtainable, and most of them can be either improved or the range of validity extended. It is shown how to generate iteratively upper and lower bounds, which are converging sequences in the case of the I-functions. Few iterations provide simple and effective upper and lower bounds for approximating the ratios Iν(x)/Iν−1(x) and the condition numbers for any x,ν?0; for the ratios Kν(x)/Kν+1(x) the same is possible, but with some restrictions on ν. Using these bounds Turán-type inequalities are established, extending the range of validity of some known inequalities and obtaining new inequalities as well; for instance, it is shown that Kν+1(x)Kν−1(x)/(Kν²(x))<|ν|/(|ν|−1), x>0, ν∉[−1,1] and that the inequality is the best possible; this proves and improves an existing conjecture.     

Multiple integrals involving product of modified Bessel functions of the second kind

The following two multiple integrals are evaluated

$$\prod\limits_{r = 1}^{m - 1} {\int\limits_0^\infty {\lambda _r^{k_r } K_{\gamma r} (\lambda _r )d\lambda _r K_\mu } \left[ {x\left( {\lambda _1 ...\lambda _{m - 1} } \right)^{ \pm 1} } \right].} $$     

Functional inequalities for modified Bessel functions

In this paper, our aim is to show some mean value inequalities for the modified Bessel functions of the first and second kind. Our proofs are based on some bounds for the logarithmic derivatives of these functions, which are in fact equivalent to the corresponding Turán-type inequalities for these functions. As an application of the results concerning the modified Bessel function of the second kind, we prove that the cumulative distribution function of the gamma–gamma distribution is log-concave. At the end of this paper, several open problems are posed, which may be of interest for further research.     

Bounds for modified bessel functions

Two-side inequalities for the modified Bessel functionI _v(x), K_v(x) of the first and third kind and of order v, are established. The chief tool is the monotonocity of the functionsI _v+1(x)/I _v(x),K _v+1(x)/K _v(x).     

Redheffer type bounds for Bessel and modified Bessel functions of the first kind

In this paper our aim is to show some new inequalities of the Redheffer type for Bessel and modified Bessel functions of the first kind. The key tools in our proofs are some classical results on the monotonicity of quotients of differentiable functions as well as on the monotonicity of quotients of two power series. We also use some known results on the quotients of Bessel and modified Bessel functions of the first kind, and by using the monotonicity of the Dirichlet eta function we prove a sharp inequality for the tangent function. At the end of the paper a conjecture is stated, which may be of interest for further research.     

Some numerical methods for approximating modified Bessel functions

In this paper the numerical Tau methods for the approximation of the kernels of Kontorovitch–Lebedev integral transforms are described. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Inequalities for modified Bessel functions and their integrals

Simple inequalities for some integrals involving the modified Bessel functions _Iν(x)$I_{\nu }(x)$ and _Kν(x)$K_{\nu }(x)$ are established. We also obtain a monotonicity result for _Kν(x)$K_{\nu }(x)$ and a new lower bound, that involves gamma functions, for _K0(x)$K_0(x)$.     

Functional inequalities involving Bessel and modified Bessel functions of the first kind

In this paper, we extend some known elementary trigonometric inequalities, and their hyperbolic analogues to Bessel and modified Bessel functions of the first kind. In order to prove our main results, we present some monotonicity and convexity properties of some functions involving Bessel and modified Bessel functions of the first kind. We also deduce some Turán and Lazarević-type inequalities for the confluent hypergeometric functions.     

Inequalities involving modified Bessel functions of the first kind II

The intrinsic properties, including logarithmic convexity (concavity), of the modified Bessel functions of the first kind and some other related functions are obtained. Several inequalities involving functions under discussion are established.     

Extension of Redheffer type inequalities to modified Bessel functions

In this note, new sharpened Redheffer type inequalities involving modified Bessel functions are established.     

Sharp Bounds for the Ratio of Modified Bessel Functions

Let \(I_{\nu }( x) \) be the modified Bessel functions of the first kind of order \(\nu \), and \(S_{p,\nu }( x) =W_{\nu }( x) ^{2}-2pW_{\nu }( x) -x^{2}\) with \(W_{\nu }( x) =xI_{\nu }( x) /I_{\nu +1}( x) \). We achieve necessary and sufficient conditions for the inequality \(S_{p,\nu }( x) <u\) or \(S_{p,\nu }( x) >l\) to hold for \(x>0\) by establishing the monotonicity of \(S_{p,\nu }(x)\) in \(x\in ( 0,\infty ) \) with \(\nu >-3/2\). In addition, the best parameters p and q are obtained to the inequality \(W_{\nu }( x) <( >) p+\sqrt{ x^{2}+q^{2}}\) for \(x>0\). Our main achievements improve some known results, and it seems to answer an open problem recently posed by Hornik and Grün (J Math Anal Appl 408:91–101, 2013).     

Two-sided bounds uniform in the real argument and the index for modified Bessel functions

Bounds uniform in the real argument and the index for the functionsa _ν (x)=xI′ _ν (x)/I′ _ν (x) andb _ν (x)=xK′ _ν (x)/K _ν (x), as well as for the modified Bessel functionsI _ν(x) andK _ν(x), are established in the quadrantx>0, ν≥0, except for some neighborhoods of the pointx=0, ν=0. Translated fromMatematicheskie Zametki, Vol. 65, No. 5, pp. 681–692, May, 1999.     

Bessel Functions: Monotonicity and Bounds

Monotonicity with respect to the order v of the magnitude ofgeneral Bessel functions C_v(x) = aJ_v(x)+bY_v(x) at positive stationarypoints of associated functions is derived. In particular, themagnitude of C_v at its positive stationary points is strictlydecreasing in v for all positive v. It follows that sup_x|J_v(x)|strictly decreases from 1 to 0 as v increases from 0 to . Themagnitude of x^1/2C_v(x) at its positive stationary points isstrictly increasing in v. It follows that sup_x|x^1/2J_v(x)| equals2/ for 0 v 1/2 and strictly increases to as v increases from1/2 to . It is shown that v^1/3sup_x|J_v(x)| strictly increases from 0 tob = 0.674885... as v increases from 0 to . Hence for all positivev and real x, where b is the best possible such constant. Furthermore, forall positive v and real x, where c = 0.7857468704... is the best possible such constant. Additionally, errors in work by Abramowitz and Stegun and byWatson are pointed out.     

Bessel functions for GL 2

In classical analytic number theory there are several trace formulas or summation formulas for modular forms that involve integral transformations of test functions against classical Bessel functions. Two prominent such are the Kuznetsov trace formula and the Voronoi summation formula. With the paradigm shift from classical automorphic forms to automorphic representations, one is led to ask whether the Bessel functions that arise in the classical summation formulas have a representation theoretic interpretation. We introduce Bessel functions for representations of GL ₂ over a finite field first to develop their formal properties and introduce the idea that the γ-factor that appears in local functional equations for L-functions should be the Mellin transform of a Bessel function. We then proceed to Bessel functions for representations of GL ₂(?) and explain their occurrence in the Voronoi summation formula from this point of view. We briefly discuss Bessel functions for GL ₂ over a p-adic field and the relation between γ-factors and Bessel functions in that context. We conclude with a brief discussion of Bessel functions for other groups and their application to the question of stability of γ-factors under highly ramified twists.