Upper Bounds for the Expected Maxima of Independent Random Variables Given Known First Four Moments

The paper is devoted to the study of conditional bounds for the expectation of the maximum of independent identically distributed standardized random variables for which the values of the skewness and kurtosis coefficients are known. With the aid of Hölder's inequality, an upper bound (in the form of a lower bound for a certain expression with parameters) is obtained and a criterion for the reachability of this estimate is formulated. A lower bound for the upper boundary of the expectation of the maximum is also found. A simpler and rougher upper bound is given in explicit form.