On Turán's first main theorem

1. I n t r o d u c t i o n . Recently, E. MAKAI 1 determined the best-possible constant in a theorem of TURAN. The theorem is quoted as TU~AN'S first main theorem, since it appeared in his book 2 as the solution of his "first problem". It will be cited in Sec. 4 below. This best-possible constant was obtained independently (though somewhat later) by the present author, whose proof appeared to be considerably shorter that the one of Mr. MAKAI. The matter now seems to have been rounded off so nicely that it is worth while to present a self-contained account of it. The arrangement is as follows. In Sec. 3 we present a new proof of an elegant theorem, due to MAKAI. ~ Sec. 4 gives the proof of the main theorem, and Sec. 6 the new proof of the fact that the constant is best-possible. Finally, Sec. 7 gives a discussion of the cases where the equality sign occurs. It should be remarked that Sec. 4 was implicitly contained in the paper of I. DANCS, 4 which inspired both Mr. MAKAI and the present author.