New sufficient conditions for p-valent functions

Abstract

Let \(\mathcal {A}_{p}\) be the class of functions f(z) of the form

$$ f(z)=z^{p}+a_{p+1}z^{p+1}+a_{p+2}z^{p+2}+\cdots , (p\in \mathbb {N}=\{1,2,3,\ldots \}) $$

that are analytic in the open unit disc \(\mathbb {U}=\big \{ z\in \mathbb {C}: |z| <1\big \}\). For \(f(z)\in \mathcal {A}_{p}\), Nunokawa considered some conditions such that f(z) is \(p-\)valent in \(\mathbb {U}\). Applying the results by Nunokawa, we discuss some interesting properties for functions \(f(z)\in \mathcal {A}_{p}\). Also, we give some examples for our results.