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We first show that both $\\int_M |S_C^-(g)|^ndV_g$ and ${\\rm vol}_g(M)$ (normalized by $S_C(g)\\ge -1$) are bounded below by $\\frac{(n\\pi)^n}{n!}\\mathrm{CanVol}(M)$ for any Hermitian metric $g$ on a compact complex $n-$manifold $M$. Here $S_C$ denotes the Chern scalar curvature, $S_C^-=\\max\\{-S_C,0\\}$ and ${\\rm CanVol}(M)$ is the canonical volume of $M$, i.e., the volume of the canonical line bundle $K_M$. 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