I need help with showing that a function f belongs to the Zygmund class. Only helpful suggestions please, no full solution. I am here to learn for myself. This is work for school (we are allowed to discuss the question and solutions with others). This is part of a single-variable calculus course.
Definition:
Let $f(x)$ be a function defined on $[-1,1]$. Then $f(x)$ is in the Zygmund class if there is a constant $C>0$ so that for $|h|<1$ (so that $f(x\pm h)$ is defined) and $x_0\in[-1,1]$ the following is fulfilled:
$\left|f(x_0+h)-2f(x_0)+f(x_0-h)\right|\leq C\left|h\right|$
Problem:
Show that if $f(x)$ is differentiable and $f'(x)$ is continuous on $[-1,1]$ then $f(x)$ is in the Zygmund class.
Thoughts:
The left side of the inequality could probably be rewritten as $f(x_0+h)-f(x)+f(x_0-h)-f(x)$ which reminds me of the definition for a derivative. If I then divide both sides by |h| and take the limit as $h \rightarrow 0$ (which I am not able to motivate) I end up with $C > 0$. I need a push in the right direction.
Suggestions?
The solution shouldn't require any knowledge about the Zygmund class itself.
Thanks
Two suggestions:
You probably know the Lipschitz class: the functions that satisfy $|f(x_0+h)-f(x)|\le L|h|$. Show that the Lipschitz condition implies the Zygmund condition.
Mean value theorem relates $f(x_0+h)-f(x)$ to the derivative of $f$. Show that a function with bounded derivative is Lipschitz.