A Polynomial Inequality Generalising an Integer Inequality

For any a := (a1, a2, . . . , an) ∈ (R), we establish inequalities between the two homogeneous polynomials ∆Pa(x, t) := (x + a1t)(x + a2t) · · · (x + ant) − x and Sa(x, y) := a1x+a2xy+ · · ·+any in the positive orthant x, y, t ∈ R. Conditions for ∆Pa(x, t) ≤ tSa(x, y) yield a new proof and broad generalization of the number theoretic inequality that for base b ≥ 2 the sum of all nonempty products of digits of any m ∈ Z never exceeds m, and equality holds exactly when all auxiliary digits are b − 1. Links with an inequality of Bernoulli are also noted. When n ≥ 2 and a is strictly positive, the surface ∆Pa(x, t) = tSa(x, y) lies between the planes y = x + tmax{ai : 1 ≤ i ≤ n − 1} and y = x + tmin{ai : 1 ≤ i ≤ n − 1}. For fixed t ∈ R, we explicitly determine functions α, β, γ, δ of a such that this surface is y = x + αt + βt2x−1 + O(x−2) as x → ∞, and y = γt+ δx+O(x) as x→ 0 + .