Conditional bounds of expected maxima of random variables and their reachability

The subject of this article is the expected maxima of an arbitrary number $n$ of independent and identically distrubuted random variables. Probability distributions with zero mean and variance of 1 and with given value of the expected maximum of $m$ independent random variables of this distribution are taken into consideration. The question of reachability of the boundaries obtained by other authors is investigated. In the cases of failure to derive the answer to this question, the obtained boundaries are specified. The problem might have various applications in queuing theory, insurance, finance, and other fields.