Question

For a₁,....,aₙ∈R, a₁<a₂<....<aₙ, show that :
n/a₁-a₀+n-1/a₂-a₁+....+
1/aₙ-aₙ₋₁≥∑^n_k=1(k²/a_k)

Answer 1

Let’s show that this inequality is incorrect. For example, let’s put n = 1, a₀ = -1, a₁ = 1 in the inequality.

The left side of inequality is equal to ½ and the right side is equal to 1. So the right side is greater than the left one, but an inequality in the question has the contrary sign.