pow:problem3f22
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| pow:problem3f22 [2022/10/09 14:04] – created mazur | pow:problem3f22 [2022/10/11 02:03] (current) – mazur | ||
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| + | <box 85% round orange| 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 {{: | ||