Raşa, Ioan Convexity properties of some entropies. (English) Zbl 1448.94118 Result. Math. 73, No. 3, Paper No. 105, 5 p. (2018). Summary: We consider a family of probability distributions depending on a real parameter \(x\), and study the logarithmic convexity of the sum of the squared probabilities. Applications concerning bounds and concavity properties of Rényi and Tsallis entropies are given. Finally, some extensions and an open problem are presented.For Part II, see [Result. Math. 74, No. 4, Paper No. 154, 9 p. (2019; Zbl 1429.26016)]. Cited in 2 ReviewsCited in 7 Documents MSC: 94A17 Measures of information, entropy 26A51 Convexity of real functions in one variable, generalizations 26D05 Inequalities for trigonometric functions and polynomials 60E05 Probability distributions: general theory Keywords:probability distribution; Rényi entropy; Tsallis entropy; log-convex function Citations:Zbl 1429.26016 PDFBibTeX XMLCite \textit{I. Raşa}, Result. Math. 73, No. 3, Paper No. 105, 5 p. (2018; Zbl 1448.94118) Full Text: DOI References: [1] Abel, U.; Gawronski, W.; Neuschel, Th, Complete monotonicity and zeros of sums of squared Baskakov functions, Appl. Math. Comput., 258, 130-137, (2015) · Zbl 1338.41017 [2] Altomare, F., Campiti, M.: Korovkin-Type Approximation Theory and Its Applications. de Gruyter, Berlin (1994) · Zbl 0924.41001 · doi:10.1515/9783110884586 [3] Bărar, A., Mocanu, G., Raşa, I.: Bounds for some entropies and special functions. Carpathian J. Math. (2018). arXiv:1801.05003v1, 8 Jan 2018 [4] Berdysheva, E., Studying Baskakov-Durrmeyer operators and quasi-interpolants via special functions, J. Approx. Theory, 149, 131-150, (2007) · Zbl 1132.41328 · doi:10.1016/j.jat.2007.04.009 [5] Gonska, H., Raşa, I., Rusu, M.-D.: Chebyshev-Grüss-type inequalities via discrete oscillations. Bul. Acad. Stiinte Repub. Mold. Mat. 1(74), 63-89 (2014). arXiv:1401.7908 [math.CA] [6] Nikolov, G., Inequalities for ultraspherical polynomials. proof of a conjecture of I. raşa, J. Math. Anal. Appl., 418, 852-860, (2014) · Zbl 1305.41017 · doi:10.1016/j.jmaa.2014.04.022 [7] Raşa, I., Entropies and Heun functions associated with positive linear operators, Appl. Math. Comput., 268, 422-431, (2015) [8] Raşa, I.: Special functions associated with positive linear operators. arXiv:1409.1015v2 (2014) [9] Raşa, I.: The index of coincidence for the binomial distribution is log-convex. arXiv:1706.05178v1, 16 Jun 2017 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.