New integral representations of nth order convex functions

AbstractIn this paper we give an integral representation of an n-convex function f in general case without additional assumptions on function f. We prove that any n-convex function can be represented as a sum of two (n+1)-times monotone functions and a polynomial of degree at most n. We obtain a decomposition of n-Wright-convex functions which generalizes and complements results of Maksa and Páles (2009) [13]. We define and study relative n-convexity of n-convex functions. We introduce a measure of n-convexity of f. We give a characterization of relative n-convexity in terms of this measure, as well as in terms of nth order distributional derivatives and Radon–Nikodym derivatives. We define, study and give a characterization of strong n-convexity of an n-convex function f in terms of its derivative f(n+1)(x) (which exists a.e.) without additional assumptions on differentiability of f. We prove that for any two n-convex functions f and g, such that f is n-convex with respect to g, the function g is the support for the function f in the sense introduced by Wąsowicz (2007) [29], up to polynomial of degree at most n

On the classes of higher-order Jensen-convex functions and Wright-convex functions

The classes of n-Wright-convex functions and n-Jensen-convex functions are compared with each other. It is shown that for any odd natural number $n$ the first one is the proper subclass of the second one. To reach this aim new tools connected with measure theory are developed

On the Raşa Inequality for Higher Order Convex Functions

We study the following (q−1)th convex ordering relation for qth convolution power of the difference of probability distributions μ and ν (ν−μ)∗q≥(q−1)cx0,q≥2, and we obtain the theorem providing a useful sufficient condition for its verification. We apply this theorem for various families of probability distributions and we obtain several inequalities related to the classical interpolation operators. In particular, taking binomial distributions, we obtain a new, very short proof of the inequality given recently by Abel and Leviatan (2020)

An application of the Choquet theorem to the study of randomly-superinvariant measures

Tyt. z nagłówka.Bibliogr. s. 325-326.Given a real valued random variable Θ we consider Borel measures μ on Β (R), which satisfy the inequality μ(B) ≥ Eμ (B-Θ) (B [formula] Β (R)) or the integral inequality [formula].We apply the Choquet theorem to obtain an integral representation of measures μ satisfying this inequality. We give integral representations of these measures in the particular cases of the random variable Θ.Dostępny również w formie drukowanej.KEYWORDS: ibackward translation operator, backward difference operator, integral inequality, extreme point

Probabilistic characterization of strong convexity

Tyt. z nagł.References p. 103.Dostępny również w formie drukowanej.ABSTRACT: Strong convexity is considered for real functions defined on a real interval. Probabilistic characterization is given and its geometrical sense is explained. Using this characterization some inequalities of Jensen-type are obtained. KEYWORDS: convexity, strong convexity, Jensen's inequality, Jensen gap of a function, distribution of a random variable, variance

Моделювальні дослідження випадкового керування доступом в безпровідній сенсорній мережі

In this paper we present the research results of WSN network work with random access control, using the PASTA system (Poisson Arrivals See Time Averages). We present the results of simulation tests of the efficiency of the network operation, for the two cases: 1) when all network nodes transmit protocols with the same average time between transmissions, 2) when the network nodes were divided into groups that have different average time between transmissions. This approach has many practical advantages which we present.В статье представлено результаты исследования работы сети БСС со случайным управлением доступом, используя систему PASTA (среднее значение за время наблюдения поступления пуассоновского потока). Приведено результаты моделирующих тестирований эффективности функционирования сети для двух случаев: 1) если все узлы сети передают протоколы с тем же средним временем между передачами, 2) если узлы сети разделены на группы, которые имеют разные средние времена между передачами. Этот подход имеет много практических преимуществ, которые отражено в статье.У статті представлено результати дослідження роботи мережі БСМ з випадковим керуванням доступом, використовуючи систему PASTA (середнє значення за час спостереження надходження пуассонівського потоку). Наведено результати моделювальних тестувань ефективності функціонування мережі для двох випадків: 1) коли всі вузли мережі передають протоколи з тим самим середнім часом між передачами, 2) якщо вузли мережі поділено на групи, які мають різні середні часи між передачами. Цей підхід має багато практичних переваг, які висвітлено у статті