Let $(M,d)$ be a separable and complete geodesic space with curvature lower bounded, by $κ\in \mathbb R$, in the sense of Alexandrov. Let $μ$ be a Borel probability measure on $M$, such that $μ\in\mathcal P_2(M)$, and that has at least one barycenter $x^{*}\in M$. We show that for any geodesically $α$-convex function $f:M\to \mathbb R$, for $α\in \mathbb R$, the inequality \[f(x^*)\le \int_M (f -\fracα{2}d^2(x^*,.))\,{\rm d}μ,\] holds provided $f$ is locally Lipschitz at $x^*$ and either positive or in $L^1(μ)$. Our proof relies on the properties of tangent cones at barycenters and on the existence of gradients for semi-concave functions in spaces with lower bounded curvature.