On the minimal norm of a non-regular generalized character of an arbitrary finite group

We prove that, for any generalized character ¿, other than a multiple of the regular character, of any finite group G, we have

Sg¿G# |¿(g)|2 = |G|/x(1) - 1,

where ¿ is an irreducible character of maximal degree of G and G# is the set of non-identity elements of G. We also determine exactly when equality is achieved.