An inequality with applications to statistical estimation for probabilistic functions of Markov processes and to a model for ecology

TLDR: In this paper, a polynomial with nonnegative coefficients homogeneous of degree d in its variables is shown to be polynomially homogeneous unless 3(3(x))>P(x), where 3(x)=x.

Abstract: 1. Summary. The object of this note is to prove the theorem below and sketch two applications, one to statistical estimation for (proba-bilistic) functions of Markov processes [l] and one to Blakley's model for ecology [4]. 2. Result. THEOREM. Let P(x)=P({xij}) be a polynomial with nonnegative coefficients homogeneous of degree d in its variables {##}. Let x= {##} be any point of the domain D: ## §:(), ]pLi ## = 1, i = l, • • • , p, j=l, • • • , q%. For x= {xij} ££> let 3(#) = 3{##} denote the point of D whose i, j coordinate is (dP\\ \\ f « dP 3(*)<i = (Xij 7—) / 2* *<i — \\ dXij\\(X)// ,-i dXij (»> Then P(3(x))>P(x) unless 3(x)=x. Notation, fi will denote a doubly indexed array of nonnegative integers: fx= {M#}> i = l> • • • > <lu i=l, • • • , A #* then denotes Ilf-iHî-i^* Similarly, c M is an abbreviation for C[ MiJ }. The polynomial P({xij}) is then written P(x) = ]CM V^-In our notation : (1) 3(&)*i = (Z) «Wnys*) / JLH CpiiijX».