Summability of a Class of Fourier-stieltjes Series

If 0 « (t)-)-s then, for any /3 > oc, Op (t)->-s. It is sufficient to assume that s = 0. By (3), since F(t) is increasing, op(t) <-a q « (t) and the required result follows immediately. t LEMMA 6. Let y(t) be an increasing function. If g(t) =10 y(u) du-sta as t-*0, then y(t)~sct a-1 The proof is similar to that of a result due to Hardy and Littlewood*, to which the lemma reduces when y(t) is a derivative. Let 0 < 0 < 1 be a fixed number ; then (1-0)ty(Ot) < g(t)-g(Ot) < (1-0)ty(t). Since g(t)-g(0t), s(1-0«)t«, we have _ a l imtl-ay(t),s 1-0' s(1-0«) lim(0t)1-«y(0t) <(1-0)0a-1' s(1-0«) i .e. lim t1-« y(t) < (1-0)0a-1 ' and the required result follows on allowing 0 to tend to 1. To complete the proof of (ii) we employ the Littlewood technique of repeated differentiation. Let F(t) = F1(t), Fk(t) = Jo Fk-1 (u) du, k = 2, 3,. .. . By Lemma 5, if 0, (t)-> s then 07, (t)-> s for an integer k, and this is equivalent to Fk(t) _ stk/k ! By repeated application of Lemma 6 we obtain F(t) st, and the desired result u(r)-s follows by Lemma 4. I am indebted to Prof. J. E. Littlewood for valuable suggestions made during the preparation of this paper .