M is a non-compact Rimannian manifold without boundary.

$f\in W_{loc}^{1,2}(M)$ satisfies $\Delta f \leq c$ in the weak sense, i.e. $$ -\int_M \langle \nabla f,\nabla \psi \rangle dvol \leq c\int_M \psi dvol $$ for any nonnegative Lipschitz function $\psi$ with compact support, where c is a constant.

Suppose in addition f is a $\lambda$-convex ($\lambda$ is a constant) function on M, i.e. for any geodesic $\gamma(t)$, $f \circ\gamma(t)-\lambda t^2/2$ is a convex function.

Then is there a constant $\mu$ such that f is $\mu$-concave? (The definition is similar as above)