The increase of sums and products dependent on (y ₁, ..., y _n ) by rearrangement of this set

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 [1] and a lemma of Duffin and Schaeffer [2].