On certain subclasses of univalent $p$-harmonic mappings

Abstract

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