Über komplementäre Ungleichungen mit drei Mittelwerten. Teil 2

In Part 1 we obtained lower and upper bounds of the expressionf(M _φ(x;α),M _ψ(y;α))?M _χ(f(x,y);α) by replacing the given sets(x)=(x ₁,...,x _n ),(y)=(y ₁,...,y _n ) by two suitably chosen sets ((u)=(u ₁,...,u _m ),(v)=(v ₁,...,v _m ), in general withm≥4. Now, in the case of upper bounds, the numberm will, under additional hypotheses, be reduced tom=3 (§ 4) and finally tom=2 (§ 5). Inequalities, complementary to the inequalities of Hölder and Minkowski and to another inequality are given as illustrations.