A matrix-approach for proving inequalities

Abstract

For special inequalities p ≤ q, where p, q are algebraic expressions such that for p and q corresponding matrices P, Q can be given, proofs can be performed by manipulating the rows of Q, such that the manipulation yields P. The paper gives an Ω with P=Ω(Q) where Ω can be seen as an algorithm in the classical sense because Ω=ω o ω o ... ω and ωis a manipulation of colums of some matrix. Two special manipulations ω ^< and ω ^〉 are presented as <-ordering and >-ordering functions. Furtheron it is shown how P, Q are to be chosen to be corresponding to p, q, i.e. mappings ϕ are given such that p=ϕ(P) and q=ϕ(Q) by example. For those ϕ-s it is shown that p=ϕ(P) ≤ ϕ(Ω(P))=ϕ(Q)=q. Although p, q need to be very special, a lot of capabilities of the introduced ϕ, ω exist, for example it can be proven algorithmically that $\sqrt[n]{{a_1 a_2 ...a_n }} \leqslant \tfrac{1}{n}\sum\nolimits_{i = 1}^n {a_i }$. Hence the perspectives of the method are, that improvements of the given ϕ, ω could give algorithms for a wider range of inequalities (for example polynomials) to be implemented in some Computer Algebra systems.