Hajek (1962) introduced a generalized $t$-statistic for testing the difference of two means of normal populations having unknown, and possibly unequal variances. While the actual distribution of this statistic was unknown, he was able to obtain bounds for the type I error. The present paper is concerned with extending Hajek's result, and obtaining bounds for the power curve, as well as type I error. The result is then a test for the Behrens-Fisher problem which guarantees that the type I error will not exceed $\alpha_0$, while, at the same time, the power against a specified alternative is at least $\beta_0$. Similar results are also obtained for the modified $t$-test, introduced by Lord (1947), in which the sample range $W$ replaces the root-mean-square $s$ as an estimate of standard deviation.
"Some Inequalities for Central and Non-Central Distributions." Ann. Math. Statist. 36 (5) 1521 - 1525, October, 1965. https://doi.org/10.1214/aoms/1177699911