Convexity Theorems

M(a, fi) S M'(ah ft)Jf^(a,, ft). This is M. Riesz's fundamental theorem. (See M. Riesz [5] and a different proof in Paley [4] ; see also a generalization of the theorem in L. C. Young [ i l ] . ) M. Riesz's argument proved the convexity only in the triangle O ^ a ^ l , 0 ^ / 3 ^ 1 , a+f3*zl. The extension to the whole quadrant is due to Thorin [9]. We shall not give the proof of the theorem here, since we intend to sketch later on the proof of a more general result. Let us only point out that if we restrict the