Remarks on a Convexity Theorem of Raşa

Alzer, Horst

doi:10.1007/s00025-020-1156-y

Cited by 6 publications

(3 citation statements)

References 5 publications

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“…For a fixed n, the convexity and even the logarithmic convexity of the function F n (x) were established in [17,27,31,32,34]. In this paper, we prove that for a fixed x, the sequence (F n (x)) n≥0 is logarithmically convex.…”

Section: Concluding Remarks and Future Workmentioning

confidence: 80%

“…Proof. It was proved (see [27,31,32]) that F n is a logarithmically convex function on [0, 1], i.e.,…”

Section: More About U N (T)mentioning

confidence: 99%

“…is connected with the Meyer-König and Zeller operators. The information potentials and the entropies associated with all these distributions were studied in [17]; see also [25][26][27]. It should be mentioned that they satisfy Heun-type differential equations: see [17].…”

Section: Introductionmentioning

confidence: 99%

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Inequalities for Information Potentials and Entropies

et al. 2020

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We consider a probability distribution p0(x),p1(x),… depending on a real parameter x. The associated information potential is S(x):=∑kpk2(x). The Rényi entropy and the Tsallis entropy of order 2 can be expressed as R(x)=−logS(x) and T(x)=1−S(x). We establish recurrence relations, inequalities and bounds for S(x), which lead immediately to similar relations, inequalities and bounds for the two entropies. We show that some sequences Rn(x)n≥0 and Tn(x)n≥0, associated with sequences of classical positive linear operators, are concave and increasing. Two conjectures are formulated involving the information potentials associated with the Durrmeyer density of probability, respectively the Bleimann–Butzer–Hahn probability distribution.

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