Abstract
Earlier investigations are extended to inequalities with three means of the formf(M ϕ (x;α),M Ψ (y;α))−M χ (f(x,y);α)≧0 (I). Replacing the given basic sets (x)=(x 1,...,x n ) and (y)=(y 1,...,y n ) by two suitably chosen sets (u)=(u 1,...,u m ) and (v)=(v 1,...,v m ), lower or upper bounds on the left side of (I) can be obtained. In the case of upper bounds these inequalities are complementary to (I). In general, the numberm is not less than 4; it may be reduced under additional hypotheses. Some examples (inequalities complementary to some additive inequalities) are given.
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Beck, E. Über komplementäre Ungleichungen mit drei Mittelwerten. Teil 1. Monatshefte für Mathematik 80, 13–29 (1975). https://doi.org/10.1007/BF01487800
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DOI: https://doi.org/10.1007/BF01487800