On certain subclasses of univalent $p$-harmonic mappings

Document Type: Research Paper

Authors

1 Department of Mathematics, Hebei University, Baoding, Hebei 071002, People's Republic of China

2 School of Science, Hebei University of Engineering, Handan, Hebei 056038, People's Republic of China

3 Department of Mathematics, Hebei University, Baoding, Hebei 071002, People's Republic of China

Abstract

In
this paper, the main aim is to introduce the class $mathcal
{U}_p(lambda,alpha,beta,k_0)$ of $p$-harmonic mappings together
with its subclasses $mathcal
{U}_p(lambda,alpha,beta,k_0)capmathcal {T}_p$ and $mathcal
{U}_p(lambda,alpha,beta,k_0)capmathcal {T}_p^0$, and
investigate the properties of the mappings in these classes. First,
we give a sufficient condition for mappings to be in $mathcal
{U}_p(lambda,alpha,beta,k_0)$ and also the characterization of
mappings in $mathcal {U}_p(lambda,alpha,beta,k_0)capmathcal
{T}_p$ for $max{0,frac{lambda-frac{1}{2}}{lambda+1}}leq
alphaleq lambda$. Second, we consider the starlikeness of
mappings in $mathcal {U}_p(lambda,alpha,beta,k_0)capmathcal
{T}_p^0$ for $max{0,frac{lambda-frac{1}{2}}{lambda+1}}leq
alphaleq lambda$. Third, extreme points of $mathcal
{U}_p(lambda,alpha,beta,k_0)capmathcal {T}_p$ for
$max{0,frac{lambda-frac{1}{2}}{lambda+1}}leq alphaleq
lambda$ are found. The support points of $mathcal
{U}_p(lambda,alpha,beta,k_0)capmathcal {T}_p$ for
$max{0,frac{lambda-frac{1}{2}}{lambda+1}}leq alphaleq
lambda$ and convolution of mappings in $mathcal
{U}_p(lambda,alpha,beta,k_0)capmathcal {T}_p$ for
$max{0,frac{lambda-frac{1}{2}}{lambda+1}}leq alphaleq
lambda$ are also discussed.

Keywords

Main Subjects