New sufficient conditions for p-valent functions

Let $$\mathcal {A}_{p}$$ A p be the class of functions f(z) of the form $$ f(z)=z^{p}+a_{p+1}z^{p+1}+a_{p+2}z^{p+2}+\cdots , (p\in \mathbb {N}=\{1,2,3,\ldots \}) $$ f ( z ) = z p + a p + 1 z p + 1 + a p + 2 z p + 2 + ⋯ , ( p ∈ N = { 1 , 2 , 3 , … } ) that are analytic in the open unit disc $$\mathbb {U}=\big \{ z\in \mathbb {C}: |z| <1\big \}$$ U = { z ∈ C : | z | < 1 } . For $$f(z)\in \mathcal {A}_{p}$$ f ( z ) ∈ A p , Nunokawa considered some conditions such that f(z) is $$p-$$ p - valent in $$\mathbb {U}$$ U . Applying the results by Nunokawa, we discuss some interesting properties for functions $$f(z)\in \mathcal {A}_{p}$$ f ( z ) ∈ A p . Also, we give some examples for our results.