Berezin Transform, Mellin Transform and Toeplitz Operators

We consider functions of the form f(1)(g)over bar(1) + h in the range of the Berezin transform B, where f(1) and g(1) are holomorphic on the unit disk D, and h is either harmonic or of the form f(2) (g)over bar(2) for some holomorphic functions f(2) and g(2) on D. First, by using the Mellin transform, we complement Ahern's Theorem (Ahern in J Funct Anal 215: 206-216, 2004) by proving that if u is an element of L-1 and B( u) is harmonic, then u is harmonic. Secondly, we extend Ahern's Theorem when h is harmonic, and give very precise relations between f1 and f2, g1 and g2 when h = f(2)(g)over bar(2) and g(2)(z) = z(n) with n >= 1. Finally, some applications of our results to the theory of Toeplitz operators are discussed.