MathDB

Problem 4

Part of 2007 VJIMC

Problems(2)

weak convexity, prove inequality

Source: VJIMC 2007 1.4

6/24/2021
Let f:[0,1]→[0,∞)f:[0,1]\to[0,\infty) be an arbitrary function satisfying f(x)+f(y)2≤f(x+y2)+1\frac{f(x)+f(y)}2\le f\left(\frac{x+y}2\right)+1 for all pairs x,y∈[0,1]x,y\in[0,1]. Prove that for all 0≤u<v<w≤10\le u<v<w\le1, w−vw−uf(u)+v−uw−uf(w)≤f(v)+2.\frac{w-v}{w-u}f(u)+\frac{v-u}{w-u}f(w)\le f(v)+2.
inequalitiesfunction
A in F, A subset B subset S implies B in F

Source: VJIMC 2007 2.4

6/24/2021
Let SS be a finite set with n elements and F\mathcal F a family of subsets of SS with the following property: A∈F,A⊆B⊆S  ⟹  B∈F.A\in\mathcal F,A\subseteq B\subseteq S\implies B\in\mathcal F.Prove that the function f:[0,1]→Rf:[0,1]\to\mathbb R given by f(t):=∑A∈Ft∣A∣(1−t)∣S∖A∣f(t):=\sum_{A\in\mathcal F}t^{|A|}(1-t)^|S\setminus A|is nondecreasing (∣A∣|A| denotes the number of elements of AA).
set theory