Do we get such an $f\ $?

Question

Does there exist any $f \in C[0,\pi]$ such that $\dfrac {\left \lvert \displaystyle \int_{0}^{\pi} f(x) \sin^2 x\ dx \right \rvert} {\sup\limits_{x \in [0,\pi]} |f(x)|} = \frac {\sqrt 3 \pi} {2 \sqrt 2} \ ?$ By Holder's inequality I find that for any $f \in C[0,\pi]$ we have $\dfrac {\left \lvert \displaystyle \int_{0}^{\pi} f(x) \sin^2 x\ dx \right \rvert} {\sup\limits_{x \in [0,\pi]} |f(x)|} \leq \frac {\sqrt 3 \pi} {2 \sqrt 2}.$ Can this bound be attained?