Convexity Properties of Some Entropies (II)

Raşa, Ioan

doi:10.1007/s00025-019-1081-0

Cited by 14 publications

(13 citation statements)

References 6 publications

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“…To prove (37), it suffices to combine (40) and (36). The inequalities (38) were proven in ( [31], (3.2) and (3.3)).…”

Section: Inequalities For Information Potentialsmentioning

confidence: 99%

“…Convexity properties of the information potentials and the associated entropies were presented in [31,32], but the hypothesis a n−k = a k , k = 0, 1, . .…”

Section: Remarkmentioning

confidence: 99%

“…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%

“…Remark 8. U n (0) = lim t→+∞ U n (t) = 1, (see (31)). These equalities, Theorem 7, and graphical experiments (see Figure 1) suggest that U n is convex on [0, t n ] and concave on [t n , +∞), for a suitable t n > 1.…”

Section: More About U N (T)mentioning

confidence: 99%

“…It would be interesting to have a proof for this shape of U n (t), and to find the value of t n . In order to compute U n (x), we have the explicit expressions (16) and (31), and the three terms of recurrence relation (29). In what follows, we provide two terms of recurrence relation.…”

Section: More About U N (T)mentioning

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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“…To prove (37), it suffices to combine (40) and (36). The inequalities (38) were proven in ( [31], (3.2) and (3.3)).…”

Section: Inequalities For Information Potentialsmentioning

confidence: 99%

“…Convexity properties of the information potentials and the associated entropies were presented in [31,32], but the hypothesis a n−k = a k , k = 0, 1, . .…”

Section: Remarkmentioning

confidence: 99%

“…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%

Section: More About U N (T)mentioning

confidence: 99%

Section: More About U N (T)mentioning

confidence: 99%

See 3 more Smart Citations