Landau’s Theorem for Planar Harmonic Mappings

Estimates on Bloch constants for planar harmonic mappings

“…For harmonic mappings in U, under suitable restriction, results similar to the Bloch and Landau theorems can be obtained, as it is proven in [4,[6][7][8], however, these results are not sharp. Chen and Xiong [9] also investigated the Bloch constant of analytic functions in the unit disk.…”

“…Consequently, f is univalent in the disk U r2 with r 2 given by (5), and f (U r2 ) contains a schlicht disk U σ2 with σ 2 given by (6 With the aid of Theorem 2.2 and Lemma 1.2, if we apply the same method as in the proof of Theorem 5 in [4], we obtain the following theorem, which have improved Theorem 5 in [4]. Theorem 2.4.…”

“…To prove the univalence of f (z) in U r1 , we adopt the method used in [6]. For z 1 = z 2 in U r (0 < r < r 1 ), by Lemma 2.1, we have…”

Some new results on planar harmonic mappings

“…Moreover, this result is sharp. The Landau theorem for harmonic mappings was established first by Chen, Gauthier and Hengartner [6], and improved by Dorff and Nowak [8], Grigoryan [9], Huang [11] and Liu [14] later. The following two theorems were proved in [6].…”

Bounded harmonic mappings

“…In particular, we obtain Coefficient bounds, univalent harmonic mappings and normal families were examined in [4,5,6,9,10,12]. The existence theorems for univalent harmonic mappings with a given dilatation can be found in [9,13].…”