A generalization of Livingston's coefficient inequalities for functions with positive real part

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.