62G30,62H10

Sharp lower and upper bounds for quasiconvex moments of generalized order statistics are proven by the use of rearranged Moriguti's inequality. Even in the second moment case, the method yields improvements of known quantile and moment bounds for the expectation of order and record statistics based on independent identically distributed random variables. The bounds are attainable providing new characterizations of three-point and two-point distributions.;