Let F(u, v) be a symmetric real function defined for α<u, v<β and assume that G(u, v, w)=F(u, v)+F(u, w)-F(v, w) is decreasing in v and w for u≦min (u, v). For any set (y)=(y 1, ..., y n ), α<y i <β, given except in arrangement Σ i =1/n F(y i, y i+1) where y n+1=y 1) is maximal if (and under some additional assumptions only if) (y) is arranged in circular symmetrical order. Examples are given and an additional result is proved on the product Π i =1/n [(y2 i-1 y2 i) m +α 1(y 2 i-1 y 2 i ) m-1+ ... +a m ] where a k ≧0 and where the set (y)=(y 1, .., y n ), y i ≧0 is given except in arrangement. The problems considered here arose in connection with a theorem by A. Lehman  and a lemma of Duffin and Schaeffer .
ASJC Scopus subject areas
- Mathematics (all)