On some applications of Noshiro-Warschawski’s theorem

We  apply  Noshiro-Warschawski's theorem to prove that if $f(z)=z+a_2z^2+\cdots$ is analytic in $|z| 0$, then $f(z)$ is univalent in $|z|<1$. Also, applying Ozaki's condition, we obtain several sufficient conditions for functions to be $p$-valent or $p$-valently starlike function in $|z|<1$.