pow:problem3f22 [Department of Mathematics and Statistics, Binghamton University]

Problem 3 (due on Monday, October 10)

a) Is there a function $f:\mathbb R\longrightarrow \mathbb R$ such that $\frac{f(x)+f(y)}{2}\geq f\left(\frac{x+y}{2}\right)+ \sin^2(x-y)$ for all $x,y\in \mathbb R$ ?

b) Is there a function $f:\mathbb R\longrightarrow \mathbb R$ such that $\frac{f(x)+f(y)}{2}\geq f\left(\frac{x+y}{2}\right)+ \sin|x-y|$ for all $x,y\in \mathbb R$ ?

The answer to a) is positive, for example $f(x)=4x^2$ has the property. The answer to b) is negative. We received only one solution, from Prof. Vladislaw Kargin, who solved a) and b) under additional assumption about $f$ (essentially that $f$ is continuous). For a detailed solution and some related material see the following link Solution.

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