It is proved that every regular expression of alphabetic width n, that is, with n occurrences of symbols of the alphabet, can be transformed into a deterministic finite automaton (DFA) with at most 2n2+(log2e22+o(1))nlnn states recognizing the same language (the best upper bound up to date is 2n). At the same time, it is also shown that this bound is close to optimal, namely, that there exist regular expressions of alphabetic width n over a two-symbol alphabet, such that every DFA for the same language has at least 2n2+(2+o(1))nlnn states (the previously known lower bound is 542n2). The same bounds are obtained for an intermediate problem of determinizing nondetermistic finite automata (NFA) with each state having all incoming transitions by the same symbol.