Some inequalities for the weighted log canonical thresholds

Let $\phi$ be a plurisubharmonic function defined in a neighborhood of the origin in ${\mathbb{C}}^n$. For each real number $t>-n$, we associate to $\phi$ the weighted log canonical threshold $$c_t(\phi ):=\sup \{c\ge 0:\mathrm{\parallel }z{\mathrm{\parallel }}^{2t}{e}^{-2c\phi }\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1\text{ near }0\}\mathrm{.}$$ In this paper, we prove a sharp slope inequality showing that all difference quotients of the function $t\mapsto c_t(\phi )$ are uniformly controlled by the Lelong number ${\nu }_{\phi }(0)$. Moreover, we derive explicit lower bounds for the growth of $c_t(\phi )$ in terms of the complex Monge-Ampère mass of $\phi$ at the origin. Our arguments combine weighted integrability estimates, restrictions to complex lines, and techniques from pluripotential theory.