# On certain subclasses of univalent $p$-harmonic mappings

Document Type : Research Paper

Authors

1 Department of Mathematics, Hebei University, Baoding, Hebei 071002, People&amp;#039;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&#039;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

### History

• Receive Date: 22 January 2013
• Revise Date: 08 March 2014
• Accept Date: 13 March 2014
• First Publish Date: 01 April 2015