THE ROPER-SUFFRIDGE EXTENSION OPERATORS ON THE CLASS OF STRONG AND ALMOST SPIRALLIKE MAPPINGS OF TYPE $beta$ AND ORDER $alpha$

Let$mathbb{C}^n$ be the space of $n$ complex variables. Let$Omega_{n,p_2,ldots,p_n}$ be a complete Reinhardt on$mathbb{C}^n$. The Minkowski functional on complete Reinhardt$Omega_{n,p_2,ldots,p_n}$ is denoted by $rho(z)$. The concept ofspirallike mapping of type $beta$ and order $alpha$ is defined.So, the concept of the strong and almost spirallike mappings of type$beta$ and order $alpha$ is discussed in this paper. From theSchwarz-Pick lemma, under certain conditions, we obtain that thegeneralized Roper-Suffridge operators preserve strong and almostspirallikeness of type $beta$ and order $alpha$ on bounded andcomplete Reinhardt domains $Omega_{n,p_1,cdots,p_n}$. For specificvalues for $alpha$ and $beta$, we obtain the correspondingdefinitions of strong spirallike mappings of type $beta$, strongand almost starlike mappings of order $alpha$, strong starlikemappings. Therefore we obtain the generalized Roper-Suffridgeoperators preserve strong spirallikeness of type $beta$, strong andalmost starlikeness of order $alpha$, strong starlikeness on thecorresponding domains. In particular, our results reduce to manywell-known results.