Convex order for convolution polynomials of Borel measures

Abstract

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]).

On the Raşa Inequality for Higher Order Convex Functions

“…During the Conference on Ulam's Type Stability (Rytro, Poland, 2014), Raşa [14] recalled his problem. Theorem 1 affirms the conjecture (see also [1][2][3][4]7,8,15] for further results on the I. Raşa problem).…”

“…[6]). Note, that in [8], we gave also necessary and sufficient condition for verification that μ and ν satisfy (3). Recently, Abel and Leviatan [2] gave a generalization of the Raşa inequality (1) to q-monotone functions.…”