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?