A solution to the problem of Raşa connected with Bernstein polynomials

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• 作者：Jacek Mrowiec ^{jmrowiec@ath.bielsko.pl" class="auth_mail" title="E-mail the corresponding author} ; Teresa Rajba ^{trajba@ath.bielsko.pl" class="auth_mail" title="E-mail the corresponding author} ; Szymon Wąsowicz ; ^{swasowicz@ath.bielsko.pl" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Bernstein polynomials ; Stochastic ordering ; Ohlin's Lemma ; Concentration inequality ; Convex functions ; Bernoulli and binomial distributions
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 February 2017
• 年：2017
• 卷：446
• 期：1
• 页码：864-878
• 全文大小：360 K

文摘

During the Conference on Ulam's Type Stability (Rytro, Poland, 2014), Ioan Raşa recalled his 25-years-old problem concerning some inequality involving the Bernstein polynomials. We offer the complete solution (in positive). As a tool we use stochastic orderings (which we prove for binomial distributions) as well as so-called concentration inequality. Our methods allow us to pose (and solve) the extended version of the problem in question.