On some Schwarz type inequalities

Abstract

First, we establish some Schwarz type inequalities for mappings with bounded Laplacian, then we obtain boundary versions of the Schwarz lemma.

The Boundary Schwarz lemma for harmonic and pluriharmonic mappings and some generalisations

“…In particular we will use here [Proposition 4.3 [19]] which is a corollary of the estimate obtained in [24] (cf. also [22]), stated here as Proposition 2.4.…”

“…Recently, these ideas were discussed at the Belgrade Analysis Seminar, and several recent results in this subject were obtained by the first author and some of his associates: M.Svetlik, A. Khalfallah, M. Mhamdi, B. Purtić, H.P. Li and the second author of this paper, see ( [15], [24], [22]). For more details see the introduction of paper [21] by the first author of this paper.…”

Upper bound of Hankel determinant for a class of analytic functions

“…Some of these studies, which are called the boundary version of Schwarz Lemma, are about estimating from below the modulus of the derivative of the function at some boundary point of the unit disc. The boundary version of Schwarz Lemma is given as follows [13]: Inequality (1.5) and its generalizations have important applications in geometric theory of functions and they are still hot topics in the mathematics literature [1][2][3][4][7][8][9][10][11][12][13]. Mercer has considered some Schwarz and Carathéodory inequalities at the boundary, as consequences of a lemma due to Rogosinski [11].…”

Schwarz-Pick Lemma for Harmonic and Hyperbolic Harmonic Functions

“…For more details and development regarding the Khavinson conjecture for harmonic functions, see [15,16,17,18,19,21,22].…”