Some cyclic and other inequalities. III

Source abstract

1. Introduction . In a recent paper (2) Daykin has made the Conjecture. If t is real, then where He has proved (1, 2) that (i) Φ( t ) is a convex function of t , (ii) Φ( t )≥ n if t ≥ 0 or 2, and (iii) Φ( t ) takes values arbitrarily close to ½ n if n is even, x 2 = x 4 = … = x n = 1, and x l = x 3 = … = x n-1 and t have suitable small positive values. It is known (3) that (1) is true if t = 1.