Abstract

Let An, Gn, Hn (respectively, A?n, G?n, H?n) denote the unweighted arithmetic, geometric, harmonic means of x1, . . . , xn (respectively, 1 - xn, . . . , 1 xj), where xj ? (0,1/2] (j = 1, . . . , n). In 1984, Wang and Wang established (Gn/G?n)n ? (An/A?n)n-1 Hn/H?n, which refines the well-known Ky Fan inequality Gn/G?n ? An/A?n. The validity of the converse inequality (Hn/H?n)n-1 An/A?n ? (Gn/G?n)n (0.1) was conjectured in 1988. In this paper we give a proof for (0.1). © Birkhäuser Verlag, 2001.