On the Raşa Inequality for Higher Order Convex Functions

We study the following $$(q-1)$$ ( q - 1 ) th convex ordering relation for qth convolution power of the difference of probability distributions $$\mu $$ μ and $$\nu $$ ν $$\begin{aligned} (\nu -\mu )^{*q}\ge _{(q-1)cx} 0 , \quad q\ge 2, \end{aligned}$$ ( ν - μ ) ∗ q ≥ ( q - 1 ) c x 0 , 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).

Word Metric, Stationary Measure and Minkowski’s Question Mark Function

Given a countably infinite group G acting on some space X, an increasing family of finite subsets Gn, x∈ X and a function f over X we consider the sums Sn(f, x) = ∑g∈Gnf(gx). The asymptotic behaviour of Sn(f, x) is a delicate problem that was studied under various settings. In the following paper we study this problem when G is a specific lattice in SL (2, ℤ ) acting on the projective line and Gn are chosen using the word metric. The asymptotic distribution is calculated and shown to be tightly connected to Minkowski’s question mark function. We proceed to show that the limit distribution is stationary with respect to a random walk on G defined by a specific measure µ. We further prove a stronger result stating that the asymptotic distribution is the limit point for any probability measure over X pushed forward by the convolution power µ∗n.

Convolution in perfect Lie groups

LetGbe a connected perfect real Lie group. We show that there exists α < dimGandp∈$\mathbb{N}$* such that if μ is a compactly supported α-Frostman Borel measure onG, then thepth convolution power μ*pis absolutely continuous with respect to the Haar measure onG, with arbitrarily smooth density. As an application, we obtain that ifA⊂Gis a Borel set with Hausdorff dimension at least α, then thep-fold product setApcontains a non-empty open set.