  For functions $p(z) = 1 + \sum_{n=1}^\infty p_n z^n$ holomorphic in the unit
disk, satisfying $ {\rm Re}\, p(z) > 0$, we generalize two inequalities proved
by Livingston in 1969 and 1985, and simplify their proofs. One of our results
states that $|p_n -w p_k p_{n-k}|\leq 2\max\{1, |1-2w|\}, w\in\mathbb{C}$.
Another result involves certain determinants whose entries are the coefficients
$p_n$. Both results are sharp. As applications we provide a simple proof of a
theorem of J.E. Brown and various inequalities for the coefficients of
holomorphic self-maps of the unit disk.
