Shapiro's Cyclic Sum

Shapiro's cyclic sum is defined by E ( x ) = ∑ i = 1 na i ( x ) , a i ( x ) = x i / ( x i + 1+ x i + 2 ) and x i = x i + n for all i ∈ ℤ . If K is the cone in R n of points with non‐negative coordinates, it is shown that the minimum of E in K is a fixed point of T 2 , where T is the non‐linear operator defined by ( Tx ) i = x n − i +1 /( x n − i +2 + x n − i +3 ) 2 for i = 1,2,…, n . It is conjectured that Tx = λ S k x , where S is the shift operator in R n , and a proof is given under some additional hypotheses. One of the consequences is a simple proof that at the minimum point, a i ( x ) = a n − i +1− k ( x ) for i = 1,2,…, n .