Let M be a hyperbolic surface and let f be a real smooth function on M.

Considering a geometric inequality which I conjectured to be true for different reasons, I get after a lot of computations that the inequality is in fact equivalent to the following

$$ \int_{M}(f^2 - \frac{1}{4}\(Delta(f))^2)dM≤ \frac{1}{V}(\int_{M}fdM)^2 $$,

where dm is the volume form on M and $V=\int_{M}dM$.

Is this inequality correct or false ?