A new criterion for 𝑝-valent functions

Abstract

In this paper we consider the classes K n + p − 1 {K_{n + p - 1}} of functions f ( z ) = z p + a p + 1 z p + 1 + ⋯ f(z) = {z^p} + {a_{p + 1}}{z^{p + 1}} + \cdots which are regular in the unit disc E = { z : | z | > 1 } E = \{ z:|z| > 1\} and satisfying the condition \[ Re ⁡ ( ( z n f ) ( n + p ) / ( z n − 1 f ) ( n + p − 1 ) ) > ( n + p ) / 2 , \operatorname {Re} \left ( {{{({z^n}f)}^{(n + p)}}/{{({z^{n - 1}}f)}^{(n + p - 1)}}} \right ) > (n + p)/2, \] where p is a positive integer and n is any integer greater than − p - p . It is proved that K n + p ⊂ K n + p − 1 {K_{n + p}} \subset {K_{n + p - 1}} . Since K 0 {K_0} is the class of p-valent functions, consequently it follows that all functions in K n + p − 1 {K_{n + p - 1}} are p-valent. We also obtain some special elements of K n + p − 1 {K_{n + p - 1}} via the Hadamard product.