A bridge connecting convex analysis and complex analysis and $L^2$-estimate of $d$ and $\bar{\mathrm{\partial }}$

We propose a way to connect complex analysis and convex analysis. As applications, we derive some results about $L^2$-estimate for $d$-equation and prove some curvature positivity related to convex analysis from well known $L^2$-estimate for $\bar{\mathrm{\partial }}$-equation or the results we prove in complex analysis.