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.     

On a Sum of Modified Bessel Functions

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 (J Multivariate Anal 6:222–236, 1976).     

Bounds for the product of modified Bessel functions

In this note our aim is to present some monotonicity properties of the product of modified Bessel functions of the first and second kind. Certain bounds for the product of modified Bessel functions of the first and second kind are also obtained. These bounds improve and extend known bounds for the product of modified Bessel functions of the 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.     

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.     

A function class of strictly positive definite and logarithmically completely monotonic functions related to the modified Bessel functions

In this paper, we give some conditions for a class of functions related to Bessel functions to be positive definite or strictly positive definite. We present some properties and relationships involving logarithmically completely monotonic functions and strictly positive definite functions. In particular, we are interested with the modified Bessel functions of the second kind. As applications, we prove the logarithmically monotonicity for a class of functions involving the modified Bessel functions of second kind and we established new inequalities for this function.     

On an identity for zeros of Bessel functions

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.     

Cross-product of Bessel functions: Monotonicity patterns and functional inequalities

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.     

Modified Dini functions: monotonicity patterns and functional inequalities

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 involving 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 kind, which was very useful in determining the radius of convexity of some normalized Bessel functions of the first kind.     

Univalence of integral operators involving Bessel functions

In this note our aim is to deduce some sufficient conditions for integral operators involving Bessel functions of the first kind to be univalent in the open unit disk. The key tools in our proofs are the generalized versions of the well-known Ahlfors’ and Becker’s univalence criteria and some inequalities for the normalized Bessel functions of the first kind.     

Complete Monotonicity of a Difference Between the Exponential and Trigamma Functions and Properties Related to a Modified Bessel Function

In the paper, the authors find necessary and sufficient conditions for a difference between the exponential function αe ^β/t , α, β > 0, and the trigamma function ψ′(t) to be completely monotonic on (0, ∞). While proving the complete monotonicity, the authors discover some properties related to the first order modified Bessel function of the first kind I ₁, including inequalities, monotonicity, unimodality, and convexity.     

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.     

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.     

Generalized Bessel functions: Theory and their applications

This paper presents 2 new classes of the Bessel functions on a compact domain [0,T] as generalized‐tempered Bessel functions of the first‐ and second‐kind which are denoted by GTBFs‐1 and GTBFs‐2. Two special cases corresponding to the GTBFs‐1 and GTBFs‐2 are considered. We first prove that these functions are as the solutions of 2 linear differential operators and then show that these operators are self‐adjoint on suitable domains. Some interesting properties of these sets of functions such as orthogonality, completeness, fractional derivatives and integrals, recursive relations, asymptotic formulas, and so on are proved in detail. Finally, these functions are performed to approximate some functions and also to solve 3 practical differential equations of fractionalorders.     

Calculation of Bessel functions by using continued fractions

We propose a new method for the calculation of Bessel functions of the first kind of integral order. By using the Laplace transformation, we solve a linear differential equation that defines the generating function for the Bessel functions expressed in terms of continued fractions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 12, pp. 1704–1705, December, 1995.     

Zeros of combinations of Bessel functions and the mean charge of graphene nanodots

We establish some properties of the zeros of sums and differences of contiguous Bessel functions of the first kind. As a by-product, we also prove that the zeros of the derivatives of Bessel functions of the first kind of different orders are interlaced the same way as the zeros of the Bessel functions themselves. As a physical motivation, we consider gated graphene nanodots subject to Berry–Mondragon boundary conditions. We determine the allowed energy levels and calculate the mean charge at zero temperature. We discuss its dependence on the gate (chemical) potential in detail and also comment on the effect of temperature.     

Studies on the zeros of Bessel functions and methods for their computation: 2. Monotonicity,convexity, concavity,and other properties

This work continues the study of real zeros of first- and second-kind Bessel functions and Bessel general functions with real variables and orders begun in the first part of this paper (see M.K. Kerimov, Comput. Math. Math. Phys. 54 (9), 1337–1388 (2014)). Some new results concerning such zeros are described and analyzed. Special attention is given to the monotonicity, convexity, and concavity of zeros with respect to their ranks and other parameters.     

Numerical solution of the Bagley–Torvik equation by the Bessel collocation method

In this article, a numerical technique is presented for the approximate solution of the Bagley–Torvik equation, which is a class of fractional differential equations. The basic idea of this method is to obtain the approximate solution in a generalized form of the Bessel functions of the first kind. For this purpose, by using the collocation points, the matrix operations and a generalization of the Bessel functions of the first kind, this technique transforms the Bagley–Torvik equation into a system of the linear algebraic equations. Hence, by solving this system, the unknown Bessel coefficients are computed. The reliability and efficiency of the proposed scheme are demonstrated by some numerical examples. Copyright © 2012 John Wiley & Sons, Ltd.     

On a remarkable sequence of Bessel matrices

A sequence of matrices whose elements are modified Bessel functions of the first kind is considered. Such a sequence arises when studying certain ordinary linear homogeneous second-order differential equations belonging to the family of double confluent Heun equations. The conjecture that these matrices are nonsingular is discussed together with its application to the problem of the existence of solutions analytic at the singular point of the equation referred to above.     

Discrete Bessel functions and partial difference equations

We introduce a new class of discrete Bessel functions and discrete modified Bessel functions of integer order. After obtaining some of their basic properties, we show that these functions lead to fundamental solutions of the discrete wave equation and discrete diffusion equation.     

Error Bounds for the Large-Argument Asymptotic Expansions of the Hankel and Bessel Functions

In this paper, we reconsider the large-argument asymptotic expansions of the Hankel, Bessel and modified Bessel functions and their derivatives. New integral representations for the remainder terms of these asymptotic expansions are found and used to obtain sharp and realistic error bounds. We also give re-expansions for these remainder terms and provide their error estimates. A detailed discussion on the sharpness of our error bounds and their relation to other results in the literature is given. The techniques used in this paper should also generalize to asymptotic expansions which arise from an application of the method of steepest descents.