On new majorization theorems

The subject of majorization is treated exten-sively, see, for instance, [1, 5] and [4] and their references. In 1947, L. Fuchs gave a weighted generalization of the well-known majorization theorem for convex functions and two sequences monotonic in the same sense, see [4, p. 419] or [6, p. 323]. For other related results, see [4, pp. 417-420] or [6, pp. 323-332]. In this paper we shall give related results in the case when only one sequence is monotonic. Moreover, while in Fuchs results we have real weights, in our results we need positive weights. These results form extensions of theorems of Kolum-ban and Mocanu [2], Toader [8] and Maligranda, Pecaric and Persson [3], as well as results from [5, pp. 337-338; 350] and [7, pp. 9 and 145-148].