On some problems concerning holomorphic and plurisubharmonic functions (Q2368630)

Let \(D\) be a domain in \(\mathbb C^n\). The author considers the problem to determine that a function \(f\) is holomorphic (pluriharmonic or \(n\)-harmonic) in \(D\) in terms of some associated function on \(D\times C\). The main results are: (1) \(f\) is holomorphic in \(D\) if and only if \(\log | p(w)-f(z)| \) is plurisubharmonic in \(D\times C\) for some holomorphic polynomial \(p(w)\). (2) \(f\) is pluriharmonic in \(D\) if and only if \(| w-f(z)| \) is plurisubharmonic in \(D\times C\). (3) Two \(n\)-harmonic functions \(f\) and \(g\) in \(D\) are holomorphic in \(D\) if and only if \(\log | w-f(z)| + \log | w-g(z)| \) is plurisubharmonic in \(D\times C\).