In this article, we introduce a 2-parameter family of affine connections and derive the Ricci curvature. We first establish an integral Bochner technique. On one hand, this technique yields a new proof to our recent work in (Li and Xia in An integral formula and its applications on sub-static manifolds, 2016) for sub-static manifolds. On the other hand, this technique leads to various geometric inequalities and eigenvalue estimates under a much more general Ricci curvature conditions. The new Ricci curvature condition interpolates between static Ricci tensor and 1-Bakry-Émery Ricci, and also includes the conformal Ricci as an intermediate case.