259 research outputs found
Tur\'an type inequalities for Kr\"atzel functions
Complete monotonicity, Laguerre and Tur\'an type inequalities are established
for the so-called Kr\"atzel function defined by
Z_{\rho}^{\nu}(u)=\int_0^{\infty}t^{\nu-1}e^{-t^{\rho}-\frac{u}{t}}\dt,
where and Moreover, we prove the complete
monotonicity of a determinant function of which entries involve the Kr\"atzel
function.Comment: 9 page
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 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 page
Bounds for Tur\'anians of modified Bessel functions
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 figure
Tur\'an type inequalities for regular Coulomb wave functions
Tur\'an, Mitrinovi\'c-Adamovi\'c and Wilker type inequalities are deduced for
regular Coulomb wave functions. The proofs are based on a Mittag-Leffler
expansion for the regular Coulomb wave function, which may be of independent
interest. Moreover, some complete monotonicity results concerning the Coulomb
zeta functions and some interlacing properties of the zeros of Coulomb wave
functions are given.Comment: 11 page
Remarks on a parameter estimation for von Mises--Fisher distributions
We point out an error in the proof of the main result of the paper of Tanabe
et al. (2007) concerning a parameter estimation for von Mises--Fisher
distributions, we correct the proof of the main result and we present a short
alternative proof.Comment: 3 page
Zeros of a cross-product of the Coulomb wave and Tricomi hypergeometric functions
Motivated by a problem related to conditions for the existence of clines in
genetics, in this note our aim is to show that the positive zeros of a
cross-product of the regular Coulomb wave function and the Tricomi
hypergeometric function are increasing with respect to the order. In
particular, this implies that the eigenvalues of a boundary value problem are
increasing with the dimension.Comment: 7 page
Landen inequalities for special functions
In this paper our aim is to present some Landen inequalities for Gaussian
hypergeometric functions, confluent hypergeometric functions, generalized
Bessel functions and for general power series. Our main results complement and
generalize some known results in the literature.Comment: 7 pages, to be published in Proc. Amer. Math. So
Functional inequalities for the Bickley function
In this paper our aim is to deduce some complete monotonicity properties and
functional inequalities for the Bickley function. The key tools in our proofs
are the classical integral inequalities, like Chebyshev, H\"older-Rogers,
Cauchy-Schwarz, Carlson and Gr\"uss inequalities, as well as the monotone form
of l'Hospital's rule. Moreover, we prove the complete monotonicity of a
determinant function of which entries involve the Bickley function.Comment: 10 page
- âŠ