We apply Noshiro--Warschawski's theorem to prove that if $f(z)=z+a^2z^2+\dots$ is analytic in $|z|<1$ and if $|\mathfrak{Re}\{zf''(z)\}|\leq\alpha|z|^\alpha$ in $|z|<1$, for some $\alpha>0$, 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 $|z|<1$.