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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 kinds are deduced. These inequalities provide sharp lower and upper bounds for the Turánian of modified Bessel functions of the first and second kinds, 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 (1932) on ordinary differential equations, an integral representation of Ismail (1977, 1990) for the quotient of modified Bessel functions of the second kind and some results of Hartman and Watson (1961, 1974, 1983). As applications of the main results some sharp Turán type inequalities are presented for the product of modified Bessel functions of the first and second kinds and it is shown that this product is strictly geometrically concave. [ABSTRACT FROM AUTHOR]

Abstract: We prove the following theorem. Let and be real numbers. The inequality holds for all positive real numbers and if and only if . Here, and denote Euler’s gamma function and Euler’s constant, respectively. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]