# Does the following inequality hold under Zygmund condition?

Let $f:R\rightarrow R$ be a continuous function which satisfies the Zygmund condition $$|f(x+\delta)+f(x-\delta)-2f(x)|\leq \text{const} \frac{|\delta|}{(\log\frac{1}{|\delta|})^{\alpha}}, \,\, \text{for any}\,\,|\delta|<1,$$ where $\alpha >1/2.$ According to the Weiss and Zygmund Theorem, $f$ is absolute continuous and $f'\in L_{p}$ for every $p>1.$

Q: For any interval $[a, c]$ such that $|[a,c]|<1$, does the following inequality hold?: $$\int_{a}^{c}\Big|\int_{a}^{x}\frac{f'(y)(y-a)dy}{(x-a)^{2}}-\int_{x}^{c}\frac{f'(y)(c-y)dy}{(c-x)^{2}}\Big|dx\leq \text{const}\frac{|[a,c]|}{(\log\frac{1}{|[a,c]|})^{\alpha}}$$