# Bernstein approximation error for functions with Lipschitz continuous derivatives

Consider a Lipschitz continuous function $$f$$ on $$[0,1]$$ with constant $$L_f$$, and its Bernstein polynomial $$B_{n,f}$$ of degree $$n$$. It can be shown that

$$\sup_{x\in[0,1]}|B_{n,f}(x) - f(x)| \leq \frac{L}{2\sqrt{n}}.$$

However, I read from somewhere that if $$f$$ has Lipschitz continuous derivative $$g$$ with constant $$L_g$$, then the bound is improved to be $$O(\frac{1}{n})$$. How is this bound derived explicitly?