Local growth of pluri-subharmonic functions

We obtain two-bound estimates for the local growth of pluri-subharmonic functions in terms of Siciak and relative extremal functions. As applications, we give simple new proofs of “Bernstein doubling inequality” and the main result in [Alexander Brudnyi, Local inequalities for pluri-subharmonic functions, Annals Math. 149 (1999), No. 2, pp. 511–533]. We propose a conjecture similar to the comparison theorem in [H. Alexander and B. A. Taylor, Comparison of two capacities in ${\mathbb{C}}^n$, Math. Z. 186 (1984), 407–417], whose validity allows to obtain bounds for the local growth of pluri-subharmonic functions solely in term of the Siciak extremal functions.