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In this paper, we focus on the generalized Marcum function of the second kind of order $$\nu >0$$ ν > 0 , defined by $$\begin{aligned} R_{\nu }(a,b)=\frac{c_{a,\nu }}{a^{\nu -1}} \int _b ^ {\infty } t^{\nu } e^{-\frac{t^2+a^2}{2}}K_{\nu -1}(at)\mathrm{d}t, \end{aligned}$$ R ν ( a , b ) = c a , ν a ν - 1 ∫ b ∞ t ν e - t 2 + a 2 2 K ν - 1 ( a t ) d t , where $$a>0, b\ge 0,$$ a > 0 , b ≥ 0 , $$K_{\nu }$$ K ν stands for the modified Bessel function of the second kind, and $$c_{a,\nu }$$ c a , ν is a constant depending on a and $$\nu $$ ν such that $$R_{\nu }(a,0)=1.$$ R ν ( a , 0 ) = 1 . Our aim is to find some new tight bounds for the generalized Marcum function of the second kind and compare them with the existing bounds. In order to deduce these bounds, we include the monotonicity properties of various functions containing modified Bessel functions of the second kind as our main tools. Moreover, we demonstrate that our bounds in some sense are the best possible ones.

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.

In this paper, we study the local zero behavior of orthogonal polynomials around an algebraic singularity, that is, when the measure of orthogonality is supported on $$ [-1,1] $$ and behaves like $$ h(x)|x - x_0|^\lambda dx $$ for some $$ x_0 \in (-1,1) $$ , where h(x) is strictly positive and analytic. We shall sharpen the theorem of Yoram Last and Barry Simon and show that the so-called fine zero spacing (which is known for $$ \lambda = 0$$ ) unravels in the general case, and the asymptotic behavior of neighbouring zeros around the singularity can be described with the zeros of the function $$ c J_{\frac{\lambda - 1}{2}}(x) + d J_{\frac{\lambda + 1}{2}}(x) $$ , where $$ J_a(x) $$ denotes the Bessel function of the first kind and order a. Moreover, using Sturm–Liouville theory, we study the behavior of this linear combination of Bessel functions, thus providing estimates for the zeros in question.

In this paper we consider some normalized Bessel, Struve and Lommel functions of the first kind, and by using the Euler-Rayleigh inequalities for the first positive zeros of combination of special functions we obtain tight lower and upper bounds for the radii of starlikeness of these functions. By considering two different normalization of Bessel and Struve functions we give some inequalities for the radii of convexity of the same functions. On the other hand, we show that the radii of univalence of some normalized Struve and Lommel functions are exactly the radii of starlikeness of the same functions. In addition, by using some ideas of Ismail and Muldoon we present some new lower and upper bounds for the zeros of derivatives of some normalized Struve and Lommel functions. The Laguerre-P\'olya class of real entire functions plays an important role in our study., Comment: 13 pages

In this paper, some geometric properties of the normalized forms of the cross-product and product of Bessel and modified Bessel functions of the first kind are studied. For the cross-product and the product, three different normalizations are investigated and for each of the six functions, the radii of starlikeness and convexity are precisely determined by using their Hadamard factorization. Necessary and sufficient conditions are also given for the parameters such that the six normalized functions are starlike in the open unit disk; however, the convex case is open for further research. The characterization of entire functions from the Laguerre–Polya class via hyperbolic polynomials plays an important role in this paper. Moreover, the interlacing properties of the zeros of the cross-product and product of Bessel functions and their derivatives are also useful in the proof of the main results.

In this study, we aimed at developing a rapid and non-destructive method to predict the genus-level taxonomic affiliation of phenotypically similar halophilic archaea of Halobacteria Class. For this purpose, representatives of three widespread and frequently isolated haloarchaeal genera (Halobacterium sp., Haloferax sp. and Halorubrum sp.) were investigated by means of Raman spectroscopy. The most relevant Raman bands exhibited at 1505, 1150 and 1000 cm−1, respectively, were ascribed to the bacterioruberin carotenoid found in haloarchaea. Due to the similar spectral profiles, a robust chemometric approach based on fuzzy principal component analysis and linear discriminant analysis for the classification of the three halophilic strains was employed, and the high-accuracy grouping of spectral data corresponding to those strains was achieved. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper the radii of starlikeness of the Jackson and Hahn-Exton $q$-Bessel functions are considered and for each of them three different normalization are applied. By applying Euler-Rayleigh inequalities for the first positive zeros of these functions tight lower and upper bounds for the radii of starlikeness of these functions are obtained. The Laguerre-P\'olya class of real entire functions plays an important role in this study. In particular, we obtain some new bounds for the first positive zero of the derivative of the classical Bessel function of the first kind.

Tight lower and upper bounds for the radius of univalence of some normalized Bessel, Struve and Lommel functions of the first kind are obtained via Euler-Rayleigh inequalities. It is shown also that the radius of univalence of the Struve functions is greater than the corresponding radius of univalence of Bessel functions. Moreover, by using the idea of Kreyszig and Todd, and Wilf it is proved that the radii of univalence of some normalized Struve and Lommel functions are exactly the radii of starlikeness of the same functions. The Laguerre-P\'olya class of entire functions plays an important role in our study., Comment: 12 pages

Some Tur\'an type inequalities for Struve functions of the first kind are deduced by using various methods developed in the case of Bessel functions of the first and second kind. New formulas, like Mittag-Leffler expansion, infinite product representation for Struve functions of the first kind, are obtained, which may be of independent interest. Moreover, some complete monotonicity results and functional inequalities are deduced for Struve functions of the second kind. These results complement naturally the known results for a particular case of Lommel functions of the first kind, and for modified Struve functions of the first and second kind., Comment: 10 pages

In this paper our aim is to study the classical Bessel-Struve kernel. Monotonicity and log-convexity properties for the Bessel-Struve kernel, and the ratio of the Bessel-Struve kernel and the Kummer confluent hypergeometric function are investigated. Moreover, lower and upper bounds are given for the Bessel-Struve kernel in terms of the exponential function and some Turan type inequalities are deduced.

In this note our aim is to present some monotonicity properties of the product of modified Bessel functions of first and second kind. Certain bounds for the product of modified Bessel functions of first and second kind are also obtained. These bounds improve and extend known bounds for the product of modified Bessel functions of first and second kind of order zero. A new Tur\'an type inequality is also given for the product of modified Bessel functions, and some open problems are stated, which may be of interest for further research., Comment: 8 pages, 1 figure

The asymptotic behaviour, with respect to the large order, of the radii of starlikeness of two types of normalised Bessel functions is considered. We derive complete asymptotic expansions for the radii of starlikeness and provide recurrence relations for the coefficients of these expansions. The proofs rely on the notion of Rayleigh sums and asymptotic inversion. The techniques employed in the paper could be useful to treat similar problems where inversion of asymptotic expansions is involved., Comment: 11 pages, revised version, accepted for publication in Journal of Mathematical Analysis and Applications. The exposition was improved based on the referee's comments

We present a detailed analysis (monotonicity, convexity, recurrence relation, closed form expression and tight bounds) of a new special function which we call as the generalized Marcum function of the second kind, which is an analogous survival (or reliability) function to the so-called generalized Marcum Q -function or the generalized Marcum function of the first kind (the survival function of the non-central chi distribution). The main difference between these two generalized Marcum functions is that they involve the modified Bessel functions of the first and second kind. In this paper we are going to see how far the properties of the generalized Marcum function of the first kind may be extended to apply to the generalized Marcum function of the second kind. These two (linearly independent) special functions have similar properties, however in case of the generalized Marcum function of the second kind we need a little bit more technical analysis in order to prove our main results.

In this paper we study the Dini functions and the cross-product of Bessel functions. Moreover, we are interested on the monotonicity patterns for the cross-product of Bessel and modified Bessel functions. In addition, we deduce Redheffer-type inequalities, and the interlacing property of the zeros of Dini functions and the cross-product of Bessel and modified Bessel functions. Bounds for logarithmic derivatives of these functions are also derived. The key tools in our proofs are some recently developed infinite product representations for Dini functions and cross-product of Bessel functions., Comment: 19 pages, one figure

The reality of the zeros of the product and cross-product of Bessel and modified Bessel functions of the first kind is studied. As a consequence the reality of the zeros of two hypergeometric polynomials is obtained together with the number of the Fourier critical points of the normalized forms of the product and cross-product of Bessel functions. As an application some geometric properties of the normalized forms of the cross-product and product of Bessel and modified Bessel functions of the first kind are studied. For the cross-product and the product three different kind of normalization are considered and for each of the six functions tight lower and upper bounds are given for the radii of starlikeness and convexity, via Euler–Rayleigh inequalities. Necessary and sufficient conditions are also given for the parameters such that four from the six normalized functions are convex in the open unit disk.

In this paper necessary and sufficient conditions are deduced for the close-to-convexity of some special combinations of Bessel functions of the first kind and their derivatives by using a result of Shah and Trimble about transcendental entire functions with univalent derivatives and some newly discovered Mittag–Leffler expansions for Bessel functions of the first kind.

In this paper some Turan type inequalities for the general Bessel function, monotonicity and bounds for its logarithmic derivative are derived. Moreover we find the series representation and the relative extrema of the Turanian of general Bessel functions. The key tools in the proofs are the recurrence relations together with some asymptotic relations for Bessel functions.

In this paper our aim is to deduce some new results on the zeros of the derivatives of some Lommel and Struve functions of the first kind and to apply these results in order to determine the radii of convexity of some normalized Lommel and Struve functions of the first kind.

The radii of \(\alpha \)-convexity are deduced for three different kinds of normalized Bessel functions of the first kind and it is shown that these radii are between the radii of starlikeness and convexity, when \(\alpha \in [0,1]\), and they are decreasing with respect to the parameter \(\alpha \). The results presented in this paper unify some recent results on the radii of starlikeness and convexity for normalized Bessel functions of the first kind. The key tools in the proofs are some interlacing properties of the zeros of some Dini functions and the zeros of Bessel functions of the first kind.

Some geometric properties of a normalized hyper-Bessel functions are investigated. Especially we focus on the radii of starlikeness, convexity, and uniform convexity of hyper-Bessel functions and we show that the obtained radii satisfy some transcendental equations. In addition, we give some bounds for the first positive zero of normalized hyper-Bessel functions, Redheffer-type inequalities, and bounds for this function. In this study we take advantage of Euler-Rayleigh inequalities and Laguerre-P\'{o}lya class of real entire functions, intensively., Comment: 12 pages

In this paper we study the inverse of the eigenfunction $\sin_p$ of the one-dimensional $p$-Laplace operator and its dependence on the parameter $p$, and we present a Tur\'an type inequality for this function. Similar inequalities are given also for other generalized inverse trigonometric and hyperbolic functions. In particular, we deduce a Tur\'an type inequality for a series considered by Ramanujan, involving the digamma function., Comment: 10 pages

Motivated by some applications in applied mathematics, biology, chemistry, physics and engineering sciences, new tight Tur\'an type inequalities for modified Bessel functions of the first and second kind are deduced. These inequalities provide sharp lower and upper bounds for the Tur\'anian 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 [19] on ordinary differential equations, an integral representation of Ismail [28,29] for the quotient of modified Bessel functions of the second kind and some results of Hartman and Watson [24,26,59]. As applications of the main results some sharp Tur\'an 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., Comment: 20 pages, 3 figures

We derive a closed formula for the determinant of the Hankel matrix whose entries are given by sums of negative powers of the zeros of the regular Coulomb wave function. This new identity applied together with results of Grommer and Chebotarev allows us to prove a Hurwitz-type theorem about the zeros of the regular Coulomb wave function. As a particular case, we obtain a new proof of the classical Hurwitz's theorem from the theory of Bessel functions that is based on algebraic arguments. In addition, several Hankel determinants with entries given by the Rayleigh function and Bernoulli numbers are also evaluated., Comment: 11 pages

In this paper our aim is to present some subordination and superordination results, by using an operator, which involves the normalized form of the generalized Bessel functions of first kind. These results are obtained by investigating some appropriate classes of admissible functions. We obtain also some sandwich-type results and we point out various known or new special cases of our main results., 15 pages, accepted in Bulletin of the Malaysian Mathematical Sciences Society

In this paper our aim is to prove some monotonicity and convexity results for the modified Struve function of the second kind by using its integral representation. Moreover, as consequences of these results, we present some functional inequalities (like Tur´an type inequalities) as well as lower and upper bounds for modified Struve function of the second kind and its logarithmic derivative.

In this paper we consider a sum of modified Bessel functions of the first kind of which particular case is used in the study of Kanter's sharp modified Bessel function bound for concentrations of some sums of independent symmetric random vectors. We present some monotonicity and convexity properties for that sum of modified Bessel functions of the first kind, as well as some Tur\'an type inequalities, lower and upper bounds. Moreover, we point out an error in Kanter's paper [Ka] and at the end of the paper we pose an open problem, which may be of interest for further research., Comment: 8 pages

Two different type integral representation formulae are established for Dini (or Fourier-Dini) series of Bessel functions. The first one is a double definite integral expression derived by using a method successfully applied in the similar questions for Neumann, Kapteyn and Schlomilch series, while the another double indefinite integral formula is concluded by means of the non-homogeneous Bessel differential equation.

Let S(α,β,λ) denote the class of analytic functions f defined on the unit disk D with the normalization f(0)=f′(0)−1=0, z/f(z)≠0 in D and satisfy the condition |f′(z)(zf(z))2−βz3(zf(z))‴−(α+β)z2(zf(z))″−1|≤λ for all z∈D and for some real constants α>−1 and β such that α+β>−1. We find conditions on constants α>−1 and β such that functions in S(α,β,λ) are univalent in D. As a consequence of our investigation, we present univalence and starlikeness criteria. As applications, we present conditions such that z/up,b,c is in S(α,β,λ), where up,b,c denotes the suitably normalized form of the generalized Bessel functions of the first kind.

We begin with a brief outline of special functions and methods, which will be needed in the next chapters. We recall here briefly the Gamma, Beta, Digamma functions, Pochhammer symbol, Bernoulli polynomials and numbers, Bessel, modified Bessel, generalized hypergeometric, Fox–Wright generalized hypergeometric, Hurwitz–Lerch Zeta functions, the Euler–Maclaurin summation formula together with Dirichlet series and Cahen’s formula, Mathieu series, Bessel and Struve differential equations, Fourier-Bessel and Dini series of Bessel functions and fractional differintegral.

In this short research note we obtain double definite integral expressions for the Kapteyn type series built by Kummer's $M$ (or confluent hypergeometric ${}_1F_1$) functions. These kind of series unify in natural way the similar fashion results for Neumann-, Schl\"omilch- and Kapteyn-Bessel series recently established by Pog\'any, S\"uli, Baricz and Jankov Ma\v{s}irevi\'c.

Recently, Pogany and Suli [Integral representation for Neumann series of Bessel functions, Proc. Amer. Math. Soc. 137(7) (2009), pp. 2363–2368] derived a closed-form integral expression for a Neumann series of Bessel functions. In this note, our aim is to establish another kind of integral representations for the Neumann series of Bessel functions of the first kind J ν.

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 Tur�n type inequalities for these functions. Moreover, we present some new Tur�n 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 FitzHugh-Nagumo equation in two spatial dimensions. At the end of this note an open problem is posed, which may be of interest for further research. � 2012 American Mathematical Society.

In this paper our aim is to show some mean value inequalities for the modified Bessel functions of the first and second kinds. Our proofs are based on some bounds for the logarithmic derivatives of these functions, which are in fact equivalent to the corresponding Tur\'an 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., Comment: 14 pages

In this paper we consider the generalized Marcum Q -function of order ν > 0 real, defined by Q ν ( a , b ) = 1 a ν - 1 ∫ b ∞ t ν e - t 2 + a 2 2 I ν - 1 ( at ) dt , where a > 0, b ⩾ 0 and I ν stands for the modified Bessel function of the first kind. Our aim is to extend some results on the (first order) Marcum Q -function to the generalized Marcum Q -function in order to deduce some new lower and upper bounds. Moreover, we show that the proposed bounds are very tight for the generalized Marcum Q -function of integer order, and we deduce some new inequalities for the more general case of real order. The chief tools in our proofs are some monotonicity properties of certain functions involving the modified Bessel function of the first kind, which are based on a criterion for the monotonicity of the quotient of two Maclaurin series.

Some new inequalities for quotients of modified Bessel functions of the first and second kinds are deduced. Moreover, some developments on bounds for modified Bessel functions of the first and second kinds, higher-order monotonicity properties of these functions and applications to a special function that arises in finite elasticity, are summarized. The key tool in our proofs is a frequently used criterion for the monotonicity of the quotient of two Maclaurin series.

In this paper our aim is to deduce some sharp Turán type inequalities for modified Bessel functions of the first and second kinds. Our proofs are based on explicit formulas for the Turánians of the modified Bessel functions of the first and second kinds and on a formula which is related to the infinite divisibility of the Studentt-distribution.

In this paper we study the generalized Marcum Q-function of order ν > 0 real, defined by Q ν ( a , b ) = 1 a ν − 1 ∫ b ∞ t ν e − t 2 + a 2 2 I ν − 1 ( a t ) dt , where a > 0 , b ⩾ 0 and I ν stands for the modified Bessel function of the first kind. Our aim is to improve and extend some recent results of Wang to the generalized Marcum Q-function in order to deduce some sharp lower and upper bounds. In both cases b ⩾ a and b a we give the best possible upper bound for Q ν ( a , b ) . The key tools in our proofs are some monotonicity properties of certain functions involving the modified Bessel function of the first kind. These monotonicity properties are deduced from some results on modified Bessel functions, which have been used in wave mechanics and finite elasticity.

Let u p be the generalized and normalized Bessel function depending on parameters b, c, p and let λ p (z) = u p (z 2), where z ∈ 𝔻 = {z ∈ ℂ : |z| q, then λ p (𝔻) ⊂ λ q (𝔻), for certain conditions on the parameters b, c, p, q. Moreover we find the radius of the smallest disk centred at 1 which contains the egg-shaped domain λ p (𝔻).

In this paper, we study the generalized Marcum Q-function Q nu(a, b), where a, nu > 0 and b ges 0. Our aim is to extend the results of Corazza and Ferrari (IEEE Trans. Inf. Theory, vol. 48, pp. 3003-3008, 2002) to the generalized Marcum Q-function in order to deduce some new tight lower and upper bounds. The key tools in our proofs are some monotonicity properties of certain functions involving the modified Bessel function of the first kind and some classical inequalities, i.e., the Cauchy-Buniakowski-Schwarz and Chebyshev integral inequalities. These bounds are shown to be very tight for large b, i.e., the relative errors of our bounds converge to zero as b increases. Both theoretical analysis and numerical results are provided to show the tightness of our bounds.

In this paper, we consider the generalized Marcum Q-function of order ν > 0 real, defined by Q ν ( a , b ) = 1 a ν - 1 ∫ b ∞ t ν e - t 2 + a 2 2 I ν - 1 ( at ) d t , where a , b ⩾ 0 , I ν stands for the modified Bessel function of the first kind and the right hand side of the above equation is replaced by its limiting value when a = 0 . Our aim is to prove that the function ν ↦ Q ν ( a , b ) is strictly increasing on ( 0 , ∞ ) for each a ⩾ 0 , b > 0 , and to deduce some interesting inequalities for the function Q ν . Moreover, we present a somewhat new viewpoint of the generalized Marcum Q-function, by showing that satisfies the new-is-better-than-used (nbu) property, which arises in economic theory.

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 Turan and Lazarevic-type inequalities for the confluent hypergeometric functions.

In this note our aim is to establish a Turán type inequality for Gaussian hypergeometric functions. This result completes the earlier result that G. Gasper proved for Jacobi polynomials. Moreover, at the end of this note we present some open problems.

We prove that for $\nu>n-1$ all zeros of the $n$th derivative of Bessel function of the first kind $J_{\nu}$ are real and simple. Moreover, we show that the positive zeros of the $n$th and $(n+1)$th derivative of Bessel function of the first kind $J_{\nu}$ are interlacing when $\nu\geq n,$ and $n$ is a natural number or zero. Our methods include the Weierstrassian representation of the $n$th derivative, properties of the Laguerre-P\'olya class of entire functions, and the Laguerre inequalities. Some similar results for the zeros of the first and second derivative of the Struve function of the first kind $\mathbf{H}_{\nu}$ are also proved. The main results obtained in this paper generalize and complement some classical results on the zeros of Bessel functions of the first kind. Some open problems related to Hurwitz theorem on the zeros of Bessel functions are also proposed, which may be of interest for further research., Comment: 10 pages

Tur\'an type inequalities for modified Bessel functions of the first kind are used to deduce some sharp lower and upper bounds for the asymptotic order parameter of the stochastic Kuramoto model. Moreover, approximation from the Lagrange inversion theorem and a rational approximation are given for the asymptotic order parameter., Comment: 10 pages, 3 figures

This paper is motivated by an open problem of Anderson, Vamanamurthy, and Vuorinen. We prove a chain of inequalities for hypergeometric functions, generalized and normalized Bessel functions, Kummer functions and for some general power series. Some particular cases and refinements are given.

Let up denote the normalized, generalized Bessel function of order p which depends on two parameters b and c and let λp(x) = up(x), x ≥ 0. It is proven that under some conditions imposed on p, b, and c the Askey inequality holds true for the function λp , i.e., that λp(x) +λp(y) ≤ 1 +λp(z), where x, y ≥ 0 and z = x + y. The lower and upper bounds for the function λp are also established.

Let aIp(x) = ∑ n 0 (x/2)2n+p n!Γ(p + an + 1) be the Galue’s generalized modified Bessel function depending on parameters a = 0, 1, 2, . . . and p > −1. Consider the function aI p : R → R, defined by aI p(x) = 2pΓ(p+1)x−paIp(x). Motivated by the inequality of Lazarevic, namely cosh x < ( sinh x x )3 for x = 0, in order to generalize this inequality we prove that the Turan-type, Lazarevic-type inequalities [aI p+1(x)] aI p(x)aI p+2(x), [aI p(x)] [aI p+1(x)] hold for all x ∈ R. Moreover, we prove that the functions p → aI p+1(x)/aI p(x), p → [aI p(x)] are increasing on (−1,∞).