The modified Lommel function $t_{μ,ν}(x)$ is an important special function, but to date there has been little progress on the problem of obtaining functional inequalities for $t_{μ,ν}(x)$. In this paper, we advance the literature substantially by obtaining a simple two-sided inequality for the ratio $t_{μ,ν}(x)/t_{μ-1,ν-1}(x)$ in terms of the ratio $I_ν(x)/I_{ν-1}(x)$ of modified Bessel functions of the first kind, thereby allowing one to exploit the extensive literature on bounds for this ratio. We apply this result to obtain two-sided inequalities for the condition numbers $xt_{μ,ν}'(x)/t_{μ,ν}(x)$, the ratio $t_{μ,ν}(x)/t_{μ,ν}(y)$ and the modified Lommel function $t_{μ,ν}(x)$ itself that are given in terms of $xI_ν'(x)/I_ν(x)$, $I_ν(x)/I_ν(y)$ and $I_ν(x)$, respectively. The bounds obtained in this paper are quite accurate and often tight in certain limits. As an important special case we deduce bounds for modified Struve functions of the first kind and their ratios, some of which are new, whilst others extend the range of validity of some results given in the recent literature.
This article studies the monotonicity, log-convexity of the modified Lommel functions by using its power series and infinite product representation. Same properties for the ratio of the modified Lommel functions with the Lommel function, $\sinh$ and $\cosh$ are also discussed. As a consequence, some Turán type and reverse Turán type inequalities are given. A Rayleigh type function for the Lommel functions are derived and as an application, we obtain the Redheffer-type inequality.
We establish some monotonicity results and functional inequalities for modified Lommel functions of the first kind. In particular, we obtain new Turán type inequalities and bounds for ratios of modified Lommel functions of the first kind, as well as the function itself. These results complement and in some cases improve on existing results, and also generalise a number of the results from the literature on monotonicity patterns and functional inequalities for the modified Struve function of the first kind.
In this paper certain Turán type inequalities for some Lommel functions of the first kind are deduced. The key tools in our proofs are the infinite product representation for these Lommel functions of the first kind, a classical result of G. Pólya on the zeros of some particular entire functions, and the connection of these Lommel functions with the so-called Laguerre-Pólya class of entire functions. Moreover, it is shown that in some cases J. Steinig's results on the sign of Lommel functions of the first kind combined with the so-called monotone form of l'Hospital's rule can be used in the pr…
In this paper, we obtain inequalities for some integrals involving the modified Lommel function of the first kind $t_{μ,ν}(x)$. In most cases, these inequalities are tight in certain limits. We also deduce a tight double inequality, involving the modified Lommel function $t_{μ,ν}(x)$, for a generalized hypergeometric function. The inequalities obtained in this paper generalise recent bounds for integrals involving the modified Struve function of the first kind.
The radii of convexity of some Lommel and Struve functions of the first kind are determined. For both of Lommel and Struve functions three different normalizations are applied in such a way that the resulting functions are analytic in the unit disk of the complex plane. Some results on the zeros of the derivatives of some Lommel and Struve functions of the first kind are also deduced, which may be of independent interest.
Simple upper and lower bounds are established for the integral $\int_0^x\mathrm{e}^{-βu}u^νt_{μ,ν}(u)\,\mathrm{d}u$, where $x>0$, $0<β<1$, $μ+ν>-2$, $μ-ν\geq-3$, and $t_{μ,ν}(x)$ is the modified Lommel function of the first kind. Our bounds complement and improve on existing bounds for this integral, by either being sharper or increasing the range of validity. Our bounds also generalise recent bounds for an integral involving the modified Struve function of the first kind, and in some cases more a direct approach lead to sharper bounds when our general bounds are specialised to the modified St…
Laurent polynomials related to the Hahn-Exton $q$-Bessel function, which are $q$-analogues of the Lommel polynomials, have been introduced by Koelink and Swarttouw. The explicit strong moment functional with respect to which the Laurent $q$-Lommel polynomials are orthogonal is given. The strong moment functional gives rise to two positive definite moment functionals. For the corresponding sets of orthogonal polynomials the orthogonality measure is determined using the three-term recurrence relation as a starting point. The relation between Chebyshev polynomials of the second kind and the Laure…
In this paper our aim is to deduce some sufficient (and necessary) conditions for the close-to-convexity of some special functions and their derivatives, like Bessel functions, Struve functions, and a particular case of Lommel functions of the first kind, which can be expressed in terms of the hypergeometric function ${}_1F_2$. The key tool in our proofs is a result of Shah and Trimble about transcendental entire functions with univalent derivatives. Moreover, a known result of Pólya on entire functions, the infinite product representations and some results on zeros of Bessel, Struve and Lomme…
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 f…
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-Pólya class of entire functions plays an important rol…
Some Turán 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 modifi…
Geometric properties of the classical Lommel and Struve functions, both of the first kind, are studied. For each of them, there different normalizations are applied in such a way that the resulting functions are analytic in the unit disc of the complex plane. For each of the six functions we determine the radius of starlikeness precisely.
Generalized integral formulas involving the generalized modified k-Bessel function $J_{k,ν}^{c,γ,λ}\left( z\right) $ of first kind are expressed in terms generalized $k-$Wright functions. Some interesting special cases of the main results are also discussed
Simple inequalities for some integrals involving the modified Struve function of the first kind $\mathbf{L}_ν(x)$ are established. In most cases, these inequalities have best possible constant. We also deduce a tight double inequality, involving the modified Struve function $\mathbf{L}_ν(x)$, for a generalized hypergeometric function.
We obtain new inequalities for the modified Bessel function of the second kind $K_ν$ in terms of the gamma function. These bounds follow as special cases of inequalities that we derive for the kernel of the Krätzel integral transformation.
Simple upper and lower bounds are established for the integral $\int_0^x\mathrm{e}^{-βt}t^ν\mathbf{L}_ν(t)\,\mathrm{d}t$, where $x>0$, $ν>-1$, $0<β<1$ and $\mathbf{L}_ν(x)$ is the modified Struve function of the first kind. These bounds complement and improve on existing results, through either sharper bounds or increased ranges of validity. In deriving our bounds, we obtain some monotonicity results and inequalities for products of the modified Struve function of the first kind and the modified Bessel function of the second kind $K_ν(x)$, as well as a new bound for the ratio $\mathbf{L}_ν(x)/…
In this paper our aim is to show some new inequalities of 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 use also 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 i…
Simple inequalities are established for integrals of the type $\int_0^x \mathrm{e}^{-γt} t^{-ν} \mathbf{L}_ν(t)\,\mathrm{d}t$, where $x>0$, $0\leqγ<1$, $ν>-\frac{3}{2}$ and $\mathbf{L}_ν(x)$ is the modified Struve function of the first kind. In most cases, these inequalities are tight in certain limits. As a consequence we deduce a tight double inequality, involving the modified Struve function $\mathbf{L}_ν(x)$, for a generalized hypergeometric function.
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 Turán 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.
In this note we offer some log-concavity properties of certain functions related to Bessel functions of the first kind and modified Bessel functions of the first and second kind, by solving partially a recent conjecture on the log-convexity/log-concavity properties for modified Bessel functions of the first kind and their derivatives. Moreover, we give an application of the mentioned results by extending two inequalities of van der Corput to Bessel and modified Bessel functions of the first kind. Similar inequalities are proved also for modified Bessel functions of the second kind, as well as …
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 [19] on ordinary differential equations, an integral representation of Ismail [28,29] for the qu…
Simple upper and lower bounds are obtained for the integral $\int_0^x\mathrm{e}^{-γt}t^νI_ν(t)\,\mathrm{d}t$, $x>0$, $ν>-\frac{1}{2}$, $0<γ<1$. Most of our bounds for this integral are tight as $x\rightarrow\infty$. We apply one of our inequalities to bound some expressions involving this integral. Two of these expressions appear in Stein's method for variance-gamma approximation, and our bounds will allow for a technical advancement to be made to the method.
It is known that Struve function $\mathbf H_ν$ and modified Struve function $\mathbf L_ν$ are closely connected to Bessel function of the first kind $J_ν$ and to modified Bessel function of the first kind $I_ν$ and possess representations through higher transcendental functions like generalized hypergeometric ${}_1F_2$ and Meijer $G$ function. Also, the NIST project and Wolfram formula collection contain a set of Kapteyn type series expansions for $\mathbf L_ν(x)$. In this paper firstly, we obtain various another type integral representation formulae for $\mathbf L_ν(x)$ using the technique de…
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 Turán type inequalities) as well as lower and upper bounds for modified Struve function of the second kind and its logarithmic derivative.
Simple inequalities are established for some integrals involving the modified Bessel functions of the first and second kind. In most cases these inequalities are tight in certain limits. As a consequence, we deduce a tight double inequality, involving the modified Bessel function of the first kind, for a generalized hypergeometric function. We also present some open problems that arise from this research.
We obtain a simple two-sided inequality for the ratio $\mathbf{L}_ν(x)/\mathbf{L}_{ν-1}(x)$ in terms of the ratio $I_ν(x)/I_{ν-1}(x)$, where $\mathbf{L}_ν(x)$ is the modified Struve function of the first kind and $I_ν(x)$ is the modified Bessel function of the first kind. This result allows one to use the extensive literature on bounds for $I_ν(x)/I_{ν-1}(x)$ to immediately deduce bounds for $\mathbf{L}_ν(x)/\mathbf{L}_{ν-1}(x)$. We note some consequences and obtain further bounds for $\mathbf{L}_ν(x)/\mathbf{L}_{ν-1}(x)$ by adapting techniques used to bound the ratio $I_ν(x)/I_{ν-1}(x)$. We a…
In this paper, we are interested in a Neumann-type series for modified Bessel functions of the first kind which arises in the study of Dunkl operators associated with dihedral groups and as an instance of the Laguerre semigroup constructed by Ben Said-Kobayashi-Orsted. We first revisit the particular case corresponding to the group of square-preserving symmetries for which we give two new and different proofs other than the existing ones. The first proof uses the expansion of powers in a Neumann series of Bessel functions while the second one is based on a quadratic transformation for the Gaus…
In this paper some new series and integral representations for the Turánian of modified Bessel functions of the first kind are given, which give new asymptotic expansions and tight bounds for the Turán determinant in the question. It is shown that in the case of natural and real order the Turánian can be represented in a relatively compact form, which yields a uniform upper bound for the Turán determinant for modified Bessel functions of the first kind. Our results complement and improve some of the results from the literature.
We deduce some new functional inequalities, like Turán type inequalities, Redheffer type inequalities, and a Mittag-Leffler expansion for a special combination of modified Bessel functions of the first kind, called modified Dini functions. Moreover, we show the complete monotonicity of a quotient of modified Dini functions by introducing a new continuous infinitely divisible probability distribution. The key tool in our proofs is a recently developed infinite product representation for a special combination of Bessel functions of the first, which was very useful in determining the radius of co…
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 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 rese…
In the paper, the author establishes inequalities, monotonicity, convexity, and unimodality for functions concerning the modified Bessel functions of the first kind and compute the completely monotonic degrees of differences between the exponential and trigamma functions.
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. Moreover, the interlacing properties of the real zeros of these products of Bessel functions and their derivatives are also obtained. As an application some geometric properties of the normalized forms of the cross-product and product of Bessel and m…
New index transforms, involving the real part of the modified Bessel function of the first kind as the kernel are considered. Mapping properties such as the boundedness and invertibility are investigated for these operators in the Lebesgue spaces. Inversion theorems are proved. As an interesting application, a solution of the initial value problem for the second order partial differential equation, involving the Laplacian, is obtained. It is noted, that the corresponding operators with the imaginary part of the modified Bessel function of the first kind lead to the familiar Kontorovich- Lebede…
Fractional kinetic equations are investigated in order to describe the various phenomena governed by anomalous reaction in dynamical systems with chaotic motion. Many authors have provided solutions for various families of fractional kinetic equations involving special functions. Here, in this paper, we aim at presenting solutions of certain general families of fractional kinetic equations associated with the generalized modified $k$-Bessel function of the first kind. It is also pointed out that the main results presented here are general enough to be able to be specialized to yield many known…
The modified Bessel function of the second kind $K_{iν}(x)$ of imaginary order for fixed $x>0$ possesses a countably infinite sequence of real zeros. Recently it has been shown that the $n$th zero behaves like $ν_n\sim πn/\log\,n$ as $n\to\infty$. In this note we determine a more precise estimate for the bahaviour of these zeros for large $n$ by making use of the known asymptotic expansion of $K_{iν}(x)$ for large $ν$. Numerical results are presented to illustrate the accuracy of the expansion obtained.
Simple inequalities are established for some integrals involving the modified Bessel functions of the first and second kind. In most cases, we show that we obtain the best possible constant or that our bounds are tight in certain limits. We apply these inequalities to obtain uniform bounds for several expressions involving integrals of modified Bessel functions. Such expressions occur in Stein's method for variance-gamma approximation, and the results obtained in this paper allow for technical advances in the method. We also present some open problems that arise from this research.
Expressions for the derivatives with respect to order of modified Bessel functions evaluated at integer orders and certain integral representations of associated Legendre functions with modulus argument greater than unity are used to compute derivatives of the associated Legendre functions with respect to their parameters. For the associated Legendre functions of the first and second kind, derivatives with respect to the degree are evaluated at odd-half-integer degrees, for general complex orders, and derivatives with respect to the order are evaluated at integer orders, for general complex de…
In this paper our aim is to present an elementary proof of an identity of Calogero concerning the zeros of Bessel functions of the first kind. Moreover, by using our elementary approach we present a new identity for the zeros of Bessel functions of the first kind, which in particular reduces to some other new identities. We also show that our method can be applied for the zeros of other special functions, like Struve functions of the first kind, and modified Bessel functions of the second kind.
Let $I_ν(x)$ and $K_ν(x)$ be the first and second kind modified Bessel functions. It is shown that the nullclines of the Riccati equation satisfied by $x^α Φ_{i,ν}(x)$, $i=1,2$, with $Φ_{1,ν}=I_{ν-1}(x)/I_ν(x)$ and $Φ_{2,ν}(x)=-K_{ν-1}(x)/K_ν(x)$, are bounds for $x^α Φ_{i,ν}(x)$, which are solutions with unique monotonicity properties; these bounds hold at least for $\pm α\notin (0,1)$ and $ν\ge 1/2$. Properties for the product $P_ν(x)=I_ν(x)K_ν(x)$ can be obtained as a consequence; for instance, it is shown that $P_ν(x)$ is decreasing if $ν\ge -1$ (extending the known range of this result) an…
The bounds for the ratios of first and second kind modified Bessel functions of consecutive orders are important quantities appearing in a large number of scientific applications. We obtain new bounds which are accurate in a large region of parameters and which are shaper than previous bounds. The new bounds are obtained by a qualitative analysis of the Riccati equation satisfied by these ratios. A procedure is considered in which the bounds obtained from the analysis of the Riccati equation are used to define a new function satisfying a new Riccati equation which yields sharper bounds. Simila…
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 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 a conjecture…
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 Turán 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.
The z-zeros of the modified Bessel function of the third kind K_{nu}(z), also known as modified Hankel function or Macdonald function, are considered for arbitrary complex values of the order nu. Approximate expressions for the zeros, applicable in the cases of very small or very large |nu|, are given. The behaviour of the zeros for varying |nu| or arg(nu), obtained numerically, is illustrated by means of some graphics.
Let $K_{0}$ denote the modified Bessel function of second kind and zeroth order. In this paper we will studying the function $\tildeω_{n}\left( x\right) :=\frac{\left( -x\right) ^{n}K_{0}^{\left( n\right) }\left( x\right) }{n!}$ for positive argument. The function $\tildeω_{n}$ plays an important role for the formulation of the wave equation in two spatial dimensions as a retarded potential integral equation. We will prove that the growth of the derivatives $\tildeω_{n}^{\left( m\right) }$ with respect to $n$ can be bounded by $O\left( \left( n+1\right) ^{m/2}\right) $ while for small and larg…
In this paper a necessary and sufficient condition is deduced for the close-to-convexity of a cross product of Bessel and modified Bessel functions of the first kind and their derivatives by using a result of Shah and Trimble about transcendental entire functions with univalent derivatives, the newly discovered power series and infinite product representation of this cross-product, as well as a slightly modified version of a result of Lorch on the monotonicity of the zeros of the cross product with respect to the order.
In 1946, Magnus presented an addition theorem for the confluent hypergeometric function of the second kind $U$ with argument $x+y$ expressed as an integral of a product of two $U$'s, one with argument $x$ and another with argument $y$. We take advantage of recently obtained asymptotics for $U$ with large complex first parameter to determine a domain of convergence for Magnus' result. Using well-known specializations of $U$, we obtain corresponding integral addition theorems with precise domains of convergence for modified parabolic cylinder functions, and Hankel, Macdonald, and Bessel function…
The $ν$-zeros of the Bessel functions of purely imaginary order are examined for fixed argument $x>0$. In the case of the modified Bessel function of the second kind $K_{iν}(x)$, it is known that it possesses a countably infinite sequence of real $ν$-zeros described by $ν_n\sim πn/\log\,n$ as $n\to\infty$. Here we apply a unified approach to determine asymptotic estimates of the $ν$-zeros of the modified Bessel functions $L_{iν}(x)\equiv I_{iν}(x)+I_{-iν}(x)$ and $K_{iν}(x)$ and the ordinary Bessel functions $J_{iν}(x)\pm J_{-iν}(x)$.
This paper is an attempt to solve an important class of hypersingular integral equations of the second kind. To this end, we apply a new weighted and modified perturbation method which includes some special cases of the Adomian decomposition method. To justify the efficiency and applicability of the proposed method, we examine some examples. The principal aspects of this method are its simplicity along with fast computations.
In the paper, the authors verify the complete monotonicity of the difference $e^{1/t}-ψ'(t)$ on $(0,\infty)$, compute the completely monotonic degree and establish integral representations of the remainder of the Laurent series expansion of $e^{1/z}$, and derive an inequality which gives a lower bound for the first order modified Bessel function of the first kind.
We consider mixed type multiple orthogonal polynomials associated with a system of weight functions consisting of two vectors. One vector is defined in terms of scaled modified Bessel function of the first kind $I_μ$ and $I_{μ+1}$, the other vector is defined in terms of scaled modified Bessel function of the second kind $K_ν$ and $K_{ν+1}$. We show that the corresponding mixed type multiple orthogonal polynomials exist. For the special case that each multi-index is on or close to the diagonal, basic properties of the polynomials and their linear forms are investigated, which include explicit …
