Treffer: P-valent harmonic mappings with finite Blaschke dilatations: \(p\)-valent harmonic mappings with finite Blaschke dilatations

The first part of this nice paper is mainly devoted to studying \(p\)-valent harmonic mappings defined in the complex plane. The authors obtain a number of interesting results for such mappings which are quite different from the corresponding ones for entire (analytic) functions. For instance, it is proved that there exists a \(p\)-valent harmonic function in \(\mathbb C\) which is not a harmonic polynomial and satisfies \(\lim_{z\to\infty }f(z)=\infty \). Also, the authors show that if the number of zeros of a harmonic polynomial of degree \(n\) is finite, then it lies between \(n\) and \(n^2\). In the second part of the paper the authors consider a Jordan domain \(D\) in \(\mathbb C\) and a finite Blaschke product \(a\) and obtain several conditions which are necessary and sufficient for the existence of a sense preserving continuous boundary correspondence \(h\) from the unit circle onto \(\partial D\) covering it \(p\) times such that its Poisson integral \(f\) is p-valent and has the second dilation \(a\).