On the bounds for the expected maxima of random samples with known expected maxima of two samples of smaller size

Our aim in the present paper is to give a new representation of the previously known estimates and further investigate upper and lower bounds for expected maxima of $n$ independent and identically distributed (i.i.d.) standardized random variables (r.v.'s) from known expected maxima of $m$ and $p$ r.v.'s with the same distribution, where $1<m<p<n$. A new representation is obtained from expansion of the inverse distribution function in a system of orthonormal functions on the unit interval. A criterion for attainability of the resulting bounds is put forward. We also obtain asymptotic properties of normed bounds for maxima expectations and normed maxima of r.v.'s with distribution where a criterion for attainability of these bound holds.