On certain subclasses of univalent $p$-harmonic mappings

Inthis paper, the main aim is to introduce the class $mathcal{U}_p(lambda,alpha,beta,k_0)$ of $p$-harmonic mappings togetherwith its subclasses $mathcal{U}_p(lambda,alpha,beta,k_0)capmathcal {T}_p$ and $mathcal{U}_p(lambda,alpha,beta,k_0)capmathcal {T}_p^0$, andinvestigate 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 ofmappings in $mathcal {U}_p(lambda,alpha,beta,k_0)capmathcal{T}_p$ for $max{0,frac{lambda-frac{1}{2}}{lambda+1}}leqalphaleq lambda$. Second, we consider the starlikeness ofmappings in $mathcal {U}_p(lambda,alpha,beta,k_0)capmathcal{T}_p^0$ for $max{0,frac{lambda-frac{1}{2}}{lambda+1}}leqalphaleq lambda$. Third, extreme points of $mathcal{U}_p(lambda,alpha,beta,k_0)capmathcal {T}_p$ for$max{0,frac{lambda-frac{1}{2}}{lambda+1}}leq alphaleqlambda$ are found. The support points of $mathcal{U}_p(lambda,alpha,beta,k_0)capmathcal {T}_p$ for$max{0,frac{lambda-frac{1}{2}}{lambda+1}}leq alphaleqlambda$ and convolution of mappings in $mathcal{U}_p(lambda,alpha,beta,k_0)capmathcal {T}_p$ for$max{0,frac{lambda-frac{1}{2}}{lambda+1}}leq alphaleqlambda$ are also discussed.