Schweitzer Mathematical Competition 2025 — Problem 9

Prompt

Let λ, μ ∈ (0,1) be constants, let I ⊆ ℝ be a nonempty open interval, let f: I → ℝ be a nonconstant lower semicontinuous function, and let p: I → (0,∞) be an arbitrary positive function. Prove that the inequality f((λp(u)u + (1−λ)p(v)v)/(λp(u) + (1−λ)p(v))) ≤ (μp(u)f(u) + (1−μ)p(v)f(v))/(μp(u) + (1−μ)p(v)) holds for every u,v ∈ I if and only if f is convex on I and λ = μ. Provide a complete, rigorous mathematical proof of both directions. Justify every nontrivial step. If your proof uses a nontrivial external theorem, state it precisely and verify its hypotheses. Do not consult external sources.