Let $f$ be an absolutely continuous, periodic with period 1 and satisfies the condition 
$$
|f(x+\delta)+f(x-\delta)-2f(x)|\leq \text{const}\frac{\delta}{(\log\frac{1}{\delta})^{\epsilon}}, \,\,\,\delta\in (0,1), \,\,\epsilon\in (0, 1]. 
$$
My question is that for any $I=[a, b],$ which is $|I|<1$ the following inequality 
\begin{equation}
\int\_{a}^{b}\Big|\frac{f(x)-f(a)}{x-a}-\frac{f(b)-f(x)}{b-x}\Big|dx\leq \text{const}\frac{|I|}{(\log\frac{1}{|I|})^{\epsilon}}
\end{equation}
is true? If it is how can I show?



A question which belongs to a class of Zygmund functions
The standard Zygmund class (with $\epsilon=0$) is the Besov space
$
B^1\_{\infty,\infty},
$
that is using a Littlewood-Paley decomposition
$$
1=\sum\_{\nu\ge 0}\varphi\_\nu(\xi),\quad \varphi\_\nu(\xi) =\phi(\vert\xi\vert 2^{-\nu}) \text{ for $\nu\ge 1$, supp$\phi=[r,R], r>0,\ \varphi\_0\in C^\infty\_c$,}
$$
$$
f\in B^1\_{\infty,\infty} \text{ means}\ \sup\_\nu2^\nu\Vert \varphi\_\nu(D)u\Vert\_{L^\infty}<+\infty.
$$
I guess that your class of functions is characterized by
$
\sup\_\nu\nu^\epsilon2^\nu\Vert \varphi\_\nu(D)u\Vert\_{L^\infty}<+\infty
$
and hope it can help for your problem by generalizing what is known for the true Zygmund class.


