Suppose a Holder function $f(x)$ satisfies $$|f(x)-f(y)|\leq |x-y|^{\alpha},\forall x,y\in [a,b]$$
where $\alpha\in (0,1]$. 
$\phi$ is a linear function satifying $$\phi(a) = f(a),\phi(b) = f(b)$$

Is it true that$|f(x)-\phi(x)|\leq |b-a|^{\alpha}, \forall x\in [a,b]$?