For any Riemannian 2-sphere $(S^2,g)$, $R$ is its scalar curvature, and $R_A$ is the average value of $R$. $f$ is the first eigenfunction of Laplacian. Then I guess that for $$ \int_{S^2} (R-R_A) f^2 \ge 0 ~~~~~~\text{and}~~~~~~ \int_{S^2} (R-R_A) |\nabla f|^2 \ge 0 $$ at least one of them is right.
Of course, I don't know how to prove it. Maybe it's not easy to prove either. If so, I want to know, what books or papers should I read if I want to study this kind of problem ?