Home > Analysis, Olympiad > Traian Lalescu contest 2008 Problem 3

Traian Lalescu contest 2008 Problem 3


Suppose U\subset \Bbb{R}^2 is a subset which contains the unit disk D, and consider a function f \in C^1(D) such that \displaystyle \left|\frac{\partial f}{\partial x}(P)\right|\leq 1, \left|\frac{\partial f}{\partial y}(P)\right|\leq 1,\ \forall P \in D.

Consider a finite set of points M_1,M_2,...,M_n \in D with the center of gravity O. Prove that for all P \in D we have the inequality \displaystyle \left|f(P)-\sum_{i=1}^n f(M_i)\right| \leq 2.

Traian Lalescu student contest 2008

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