  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.

Complex Variables