Functions with Positive Real Part

This chapter studies the class \(\mathcal {P}\) of analytic functions \(p:{\mathbb {D}}\rightarrow \mathbb {C}\) satisfying \(p(0)=1\) and \({\text {Re}}p(z)>0\) for all \(z\in \mathbb {D}\) . A function in \(\mathcal {P}\) is referred to as a function with positive real part or a Carathéodory function. Several important classes of univalent functions are closely related to this class. For instance, by the Noshiro-Warschawski theorem, a function \(f\in \mathcal {A}\) is univalent if its derivative \(f' \in \mathcal {P}\) . The class \(\mathcal {P}\) is also related to \(\mathcal {B}_0\) (see Theorem 4.4.5). Moreover, the classes of starlike and convex functions, as well as several others to be discussed later, are linked to \(\mathcal {P}\) . Distortion, growth and coefficient estimates for functions in \(\mathcal {P}\) often lead to the corresponding estimates for these related classes. In view of these connections, we examine various properties of the functions with positive real part, including growth estimates and coefficient inequalities.