Let $L\rightarrow X$ be a holomorphic line bundle generated by global sections $s_1,...,s_k$. One can define a Hermitian metric on $L$ by $$|s|^2 = \dfrac{|\psi(s)|^2}{\sum_i |\psi(s_i)|^2}$$ in a local trivialization $\psi$.
I am wondering how to prove that the curvature of this metric is positive by a straightforward computation, that is computing $$-i\partial\bar{\partial}\log(1/\sum_i |\psi(s_i)|^2)$$ and showing it is indeed a positive (1,1) form. I am getting into obscure expressions. Can someone tell me what I am supposed to get?
EDIT: So it's not actually hard: we have though for any holomorphic function $f$ $$\partial\bar{\partial}\log|f|^2 = \partial\left(\frac{\bar{\partial}|f|^2}{|f|^2} \right) =\partial\left(\frac{f\bar{\partial}\bar{f}}{|f|^2} \right) \\ =\partial\left(\frac{f}{|f|^2} \right)\wedge \bar{\partial}\bar{f}= \partial\left(\frac{1}{\bar{f}} \right)\wedge \bar{\partial}\bar{f}=0$$ which kills my expression above. Where is my mistake?