Constraints on fourth order generalized f(R) gravity

A fourth order generalized f® gravity theory (FOG) is considered with the Einstein-Hilbert action $R+a{R}^2+b{R}_{\mu \nu }{R}^{\mu \nu },$ $R_{\mu \nu }$ being Ricci'{}s tensor and R the curvature scalar. The field equations are applied to spherical bodies where Newtonian gravity is a good approximation. The result is that for $0\le a\sim -b<<R^2$, $R$ being the body radius, the gravitational field outside the body contains two Yukawas, one attractive and the other one repulsive, in addition to the Newtonian term. For $a\sim -b>>R^{2}$ the gravitational field near the body is zero but at distances greater than $\sqrt{a}\sim \sqrt{-b}$ the field is practically Newtonian. From the comparison with laboratory experiments I conclude that $\sqrt{a}$ and $\sqrt{-b}$ should be smaller than a few millimeters, which excludes any relevant effect of FOG on stars, galaxies or cosmology.