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arXiv · 1703.10634

Muirhead inequality for convex orders and a problem of I. Ra\c{s}a on Bernstein polynomials

Andrzej Komisarski·Teresa Rajba
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Abstract

We present a new, very short proof of a conjecture by I. Ra\c{s}a, which is an inequality involving basic Bernstein polynomials and convex functions. It was affirmed positively very recently by J. Mrowiec, T. Rajba and S. W\k{a}sowicz (2017) by the use of stochastic convex orders, as well as by Abel (2017) who simplified their proof. We give a useful sufficient condition for the verification of some stochastic convex order relations, which in the case of binomial distributions are equivalent to the I. Ra\c{s}a inequality. We give also the corresponding inequalities for other distributions. Our methods allow us to give some extended versions of stochastic convex orderings as well as the I. Ra\c{s}a type inequalities. In particular, we prove the Muirhead type inequality for convex orders for convolution polynomials of probability distributions.

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Andrzej Komisarski, Teresa Rajba. 2017-03-30. Muirhead inequality for convex orders and a problem of I. Ra\c{s}a on Bernstein polynomials. https://arxiv.org/abs/1703.10634

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