International Mathematics Competition
for University Students
2017

IMC2017: Day 1, Problem 2

2. Let $f\colon\mathbb{R}\to(0,\infty)$ be a differentiable function, and suppose that there exists a constant $L>0$ such that $$ \bigl|f'(x)-f'(y)\bigr| \leq L\bigl|x-y\bigr| $$ for all $x,y$. Prove that $$ \big(f'(x)\big)^2 < 2Lf(x) $$ holds for all $x$.

Proposed by: Jan Šustek, University of Ostrava

Hint: Integrate $f'$ over an interval $[x,x+\Delta]$.

  Solution