Notes on functions on the unit disk

As usual, R denotes the real numbers, Z denotes the integers, and C denotes the complex numbers. Thus each α ∈ C can be expressed as a + b i, where a, b are real numbers and i = −1. We call a, b the real and imaginary parts of α, and denote them Reα, Imα, respectively. If z is a complex number, with z = x + y i where x, y are the real and imaginary parts of z, then we write z for the complex conjugate of z, which is defined to be x − y i. The complex conjugate of a sum or product of two complex numbers is equal to the corresponding sum or product of the complex conjugates of the two complex numbers. The modulus of z is denoted |z| and is the nonnegative real number defined by |z| = x + y. This is equivalent to saying that |z| = z z. The triangle inequality for the modulus states that |z + w| ≤ |z| + |w| for all