Narrow Results

The primary objective of this paper is to explore a novel subclass ${{\mathscr{H}}{𝒮}}_p(q,\alpha )$ of $p$-harmonic mappings, along with the associated subclass ${{\mathscr{H}}{𝒮}}_p^0(q,\alpha )$. We demonstrate that the mapping ${{\mathscr{H}}{𝒮}}_p(q,\alpha )$ is both univalent and sense-preserving within the unit disk $𝕌$. Furthermore, we determine the extreme points of ${{\mathscr{H}}{𝒮}}_p^0(q,\alpha )$ and establish that ${{\mathscr{H}}{𝒮}}_p(q,\alpha )\cap {{𝒯}}_p$ is defined. Additional results include the derivation of distortion bounds, the convolution condition, and the convex combination for this subclass. Finally, we examine the class-preserving integral operator and introduce a $q$-Jackson type integral operator.

In this paper, we establish two new versions of Landau-type theorems for pluriharmonic mappings with a bounded distortion. Then using these results, we derive three Bloch-type theorems of pluriharmonic mappings, which improve the corresponding results of Chen and Gauthier.

In this paper, we introduce a new class of sense preserving harmonic mappings $f=h+\overline{g}$ in the open unit disk and prove that functions in this class are close-to-convex. We give some basic properties such as coefficient bounds, growth estimates, convolution and determine the radius of convexity for the sections of functions belonging to this family. In addition, we construct certain harmonic univalent polynomials belonging to this family.