On Block Sensitivity and Fractional Block Sensitivity

We investigate the relation between the block sensitivity $\text{bs}(f)$ and fractional block sensitivity $\text{fbs}(f)$ complexity measures of Boolean functions. While it is known that $\text{fbs}(f)=O(\text{bs}(f{)}^2)$, the best known separation achieves $$\text{fbs}$(f) = \left(\frac{1}{3\sqrt2} +o(1)\right) $\text{bs(f)}$^{3/2}$. We improve the constant factor and show a family of functions that give $$\text{fbs}$(f) = \left(\frac{1}{\sqrt6}-o(1)\right) $\text{bs}$(f)^{3/2}.$