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Autori: Árpád Baricz, Saminathan Ponnusamy, Matti Vuorinen
Editorial: Christian Berg, Elsevier, Expositiones Mathematicae, 29(4), p.399-414, 2011.
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.
Cuvinte cheie: Functional inequalities, Modified Bessel functions, Convexity with respect to Hölder means, Log-convexity, Geometrical convexity, Gamma–gamma distribution, Turán-type inequality