Some Reverses of the Generalised Triangle Inequality in Complex Inner Product Spaces

was first discovered by M. Petrovich in 1917, [5] (see [4, p. 492]) and subsequently was rediscovered by other authors, including J. Karamata [2, p. 300 – 301], H.S. Wilf [6], and in an equivalent form by M. Marden [3]. The first to consider the problem of obtaining reverses for the triangle inequality in the more general case of Hilbert and Banach spaces were J.B. Diaz and F.T. Metcalf [1] who showed that in an inner product space H over the real or complex number field, the following reverse of the triangle inequality holds (1.3) r n ∑