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

On an inequality concerning orthogonal polynomials

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Artificial earth satellites

A general discussion is given of the scientific and technological problems involved in the launching of earth satellites, covering various aspects such as the dynamical relationships involved, the propulsion and guidance systems, tracking and data transmission etc. The possible uses of satellites for scientific research as well as some of the likely future developments are also briefly indicated

Fragmentation and lethality

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An approximate method for estimating rocket trajectory elements

In this paper the problem of using composite heptatubular charges in conventional guns for a linear law of burning has been discussed. A solution of the basic equations of Internal Ballistics has been derived under certain conditions, viz., the value of the radio of specific heats, has been assumed to be the same for both the charges and the covolume correction have not been neglected