Muirhead inequality for convex orders and a problem of I. Raşa on Bernstein polynomials

Komisarski, Andrzej; Rajba, Teresa

doi:10.1016/j.jmaa.2017.09.047

Cited by 10 publications

(21 citation statements)

References 8 publications

(26 reference statements)

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“…. , 0), we get the following corollary, which generalizes Raşa type inequalities proved in [11], [7] and [5].…”

Section: The Case Of M Measuressupporting

confidence: 69%

“…if ϕ is convex and decreasing), then inequality (4.21) is valid (cf. Theorem 2.3 above and Theorem 2.6.c in [7]). Using the same method it can be shown that if u → ϕ u n+u is concave on [0, ∞) but it is not linear (e.g.…”

Section: Open Problemsmentioning

confidence: 84%

“…Note that binomially distributed random variables X ∼ B(n, x) and Y ∼ B(n, y) satisfy the condition (4.12) (see [7]…”

Section: Open Problemsmentioning

confidence: 99%

“…In [7], we considered also a generalization of (1.2), taking a finite sequence of probability distributions in place of two probability distributions µ and ν. We proved the Muirhead type inequality for convex orders for convolution polynomials, and we gave a strong generalization of Theorem 2.3.…”

Section: Introductionmentioning

confidence: 99%

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Convex order for convolution polynomials of Borel measures

Komisarski

Rajba

2019

Journal of Mathematical Analysis and Applications

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We give necessary and sufficient conditions for Borel measures to satisfy the inequality introduced by Komisarski, Rajba (2018). This inequality is a generalization of the convex order inequality for binomial distributions, which was proved by Mrowiec, Rajba, Wąsowicz (2017), as a probabilistic version of the inequality for convex functions, that was conjectured as an old open problem by I. Raşa.We present also further generalizations using convex order inequalities between convolution polynomials of finite Borel measures. We generalize recent results obtained by B. Gavrea (2018) in the discrete case to general case. We give solutions to his open problems and also formulate new problems.where µ and ν are two probability distributions on R. The inequality (1.2) can be regarded as the Raşa type inequality. In [7], we proved Theorem 2.3 providing a very useful sufficient condition for verification that µ and ν satisfy (1.2). We applied Theorem 2.3 for µ and ν from various families of probability distributions. In particular, we obtained a new proof for binomial distributions, which is significantly simpler and shorter than that given in [11]. By (1.2), we can also obtain inequalities related to some approximation operators associated with µ and ν. (such as Bernstein-Schnabl operators, Mirakyan-Szász operators, Baskakov operators and others, cf. [7]).

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“…. , 0), we get the following corollary, which generalizes Raşa type inequalities proved in [11], [7] and [5].…”

Section: The Case Of M Measuressupporting

confidence: 69%

Section: Open Problemsmentioning

confidence: 84%

“…Note that binomially distributed random variables X ∼ B(n, x) and Y ∼ B(n, y) satisfy the condition (4.12) (see [7]…”

Section: Open Problemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Convex order for convolution polynomials of Borel measures

Komisarski

Rajba

2019

Journal of Mathematical Analysis and Applications

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“…Starting from these remarks, the second author presented the inequality (4) as an open problem in [10]. A probabilistic solution was found by A. Komisarski and T. Rajba [5] using the methods developed in [8] and [6].…”

Section: Introductionmentioning

confidence: 99%