Global boundedness of functions of finite order that are bounded outside small sets

We prove that subharmonic or holomorphic functions of finite order on the plane, in space, or on the unit disc or ball that are bounded above on a sequence of circles or spheres, or on a system of embedded discs or balls, outside some asymptotically small sets are bounded above throughout. Hence, subharmonic functions of finite order on the complex plane, entire or plurisubharmonic functions of finite order, and also convex or harmonic functions of finite order that are bounded above on spheres outside such sets are constants. The results and the approaches to the proofs are new for both functions of one and several variables. Bibliography: 14 titles.

entire function of finite order, (pluri)subharmonic function, holomorphic function in the unit ball, convex function, Liouville's theorem