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)}{xa}\frac{f(b)f(x)}{bx}\Bigdx\leq \text{const}\frac{I}{(\log\frac{1}{I})^{\epsilon}} \end{equation} is true? If it is how can I show?

$\begingroup$ The Article "A Note on Smooth Functions" from Weiss and Zygmund contains also information about your class of functions. goo.gl/ha7i4l $\endgroup$ – CPJ Sep 19 '14 at 15:55
The standard Zygmund class (with $\epsilon=0$) is the Besov space $ B^1_{\infty,\infty}, $ that is using a LittlewoodPaley 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.

$\begingroup$ Thank you very much Dear Bazin. They are very useful facts, could you suggest me any literature regarding to these facts. $\endgroup$ – sokho Jan 29 '14 at 23:46

$\begingroup$ The book Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren Vol. 343 by Bahouri, Hajer, Chemin, JeanYves, Danchin, Raphaël contains plenty of information on basic Fourier analysis. $\endgroup$ – Bazin Jan 30 '14 at 8:02