Rudin PMA p.116
Let $f\colon (a,\infty) \rightarrow \mathbb{R}^k$ be a twice differentiable function $ (a\in \mathbb{R})$.
Suppose $|f|,|f'|,|f''|$ has finite upper bounds $M_0,M_1,M_2$ respectively.
How do i prove that ${M_1}^2 \leq 4M_0 M_2$?
I have proved, if $f$ is a real function, above inequality holds.
I have found that, for an arbitrary $h>0$ and $x\in (a,\infty)$,
$f(x+2h)=f(x)+2hf'(x)+ 2h^2 ({f_1}''(\xi_1), ... , {f_k}''(\xi_k))$ where $\xi_i \in (x,x+2h)$.
Does this imply the inequality above?