The one-variable star function packages a meromorphic function's value-distribution functionals into a single subharmonic function on the upper half-plane, whose boundary values are the zero and pole counting functions and whose harmonicity singles out extremal functions. Extending it to several variables by restricting to complex lines through the origin and averaging the slice star functions over the unit sphere requires knowing that the slice functionals vary continuously with the direction, which holds off a null set of directions - those meeting the indeterminacy locus of the quotient of coprime entire functions. The averaged function is again subharmonic and continuous, and it is harmonic precisely when almost every slice is, which forces a rigid global form: the function must be a one-variable canonical product with zeros on a ray and poles on the opposite ray, composed with a single complex linear form given by the gradient at the origin. Harmonicity on finitely many slices, or mere collinearity of slice zeros, is not enough.
Abi-Khuzam, Bertrand and Della Sala define a Baernstein star function for a meromorphic function of several complex variables. In one variable the star function is built by fixing a radius and an ang…