Some comments on six inequalities associated with the inefficiency of ordinary least squares with one regressor

The study of the inefficiency of the ordinary least-squares estimator (OLSE) with one regressor by Watson (1951) required a lower bound for the efficiency defined as the ratio of the variance of the best linear unbiased estimator (BLUE) to the variance of the OLSE. Such a lower bound was provided by the Cassels inequality (1951), which we note is closely related to five other inequalities, including the well-known inequality usually attributed to Kantorovich (1948), but which was established already by Frucht (1943). The main purpose in this paper is to show how these six inequalities are related, with a historical perspective. We present some proofs and conclude that all six inequalities are essentially equivalent, in the sense that any one inequality implies the other five. We identify conditions for equality in each inequality and present the six continuous integral analogues. We end the paper with English translations of the seminal papers by Frucht (1943) and Schweitzer (1914), respectively from the Spanish and Hungarian, and a fairly extensive bibliography