On starlikeness of $p$-valent analytic functions

The known Ozaki’s condition says that $\mathfrak{R}\mathfrak{e}\{f^{(p)}(z)\}>0$ for $\mathrm{\mid }z\mathrm{\mid }<1$ implies that $f(z)=z^p+a_{p+1}{z}^{p+1}+\cdots$ is at most $p$-valent in $\mathbb{D}$. In this paper prove an extension of Ozaki’s condition. Also, we shall determine the new sufficient conditions for functions to be in the class of $p$-valent starlike of order $\alpha$.