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Let $\{M_j\}$ be a sequence of positive numbers with $M_0 = 1$ (not sure this last condition is actually necessary). Define for each $j$, $$ M_j^* = \inf\{M_j, M_k^{\frac{l - j}{l - k}}M_l^{\frac{j - k}{l - k}}, \text{when } k < j < l\}. $$ Then $(M_j^*)^2 \leq M^*_{j - 1}M^*_{j + 1}$.

In his book, "The Analysis of Linear Partial Differential Operators", Hörmander claims this is true but doesn't provide a proof, even though he uses this result in the proof of the Denjoy-Carleman theorem.

Any help?

João Caminada
  • 839
  • Start by taking the log of everything, so you're trying to show $x_j^* \le \frac12 ( x_{j-1}^* + x_{j+1}^)$ for $x_j^ = \inf{ x_j, \frac{1}{l-k}((l-j)x_k + (j-k)x_l) }$, and then draw some pictures. You should find $x_j^*$ has a very natural geometric interpretation in terms of line segments joining $(k,x_k)$ to $(l,x_l)$. – Anthony Carapetis Sep 09 '17 at 05:12

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