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

Abstract

Let<br />$mathbb{C}^n$ be the space of $n$ complex variables. Let<br />$Omega_{n,p_2,ldots,p_n}$ be a complete Reinhardt on<br />$mathbb{C}^n$. The Minkowski functional on complete Reinhardt<br />$Omega_{n,p_2,ldots,p_n}$ is denoted by $rho(z)$. The concept of<br />spirallike mapping of type $beta$ and order $alpha$ is defined.<br />So, the concept of the strong and almost spirallike mappings of type<br />$beta$ and order $alpha$ is discussed in this paper. From the<br />Schwarz-Pick lemma, under certain conditions, we obtain that the<br />generalized Roper-Suffridge operators preserve strong and almost<br />spirallikeness of type $beta$ and order $alpha$ on bounded and<br />complete Reinhardt domains $Omega_{n,p_1,cdots,p_n}$. For specific<br />values for $alpha$ and $beta$, we obtain the corresponding<br />definitions of strong spirallike mappings of type $beta$, strong<br />and almost starlike mappings of order $alpha$, strong starlike<br />mappings. Therefore we obtain the generalized Roper-Suffridge<br />operators preserve strong spirallikeness of type $beta$, strong and<br />almost starlikeness of order $alpha$, strong starlikeness on the<br />corresponding domains. In particular, our results reduce to many<br />well-known results.