We propose a new class of divergence measures for Independent Component Analysis (ICA) for the demixing of multiple source mixtures. We call it the Convex Cauchy-Schwarz Divergence (CCS-DIV), and it is formed by integrating convex functions into the Cauchy-Schwarz inequality. The new measure is symmetric and the degree of its curvature with respect to the joint-distribution can be tuned by a (convexity) parameter. The CCS-DIV is able to speed-up the search process in the parameter space and produces improved demixing performance. An algorithm, generated from the proposed divergence, is developed which is employing the non-parametric Parzen window-based distribution. Simulation evidence is presented to verify and quantify its superior performance in comparison to state-of-the art approaches.