Let $f \in C^n(\mathbb{ R})$ be such that $\Vert f^{ (n)} \Vert_\infty < \infty$. Let $f^{ [n]} \in C(\mathbb{ R}^{ n+1})$ be the $n$th order divided difference. A special case of our main result states that for $1 < p < \infty$ we have \[\Vert T_{ f^{ [n]}}: S_{ np} \times \ldots \times S_{ np} \rightarrow S_{ p} \Vert \lesssim p^\ast p^n \Vert f^{ (n)} \Vert_\infty, \] where $p^\ast = p/(p-1)$ is the H\"older conjugate of $p$ and $T_{ f^{ [n]}}$ is the multilinear Schur multiplier with symbol $f^{ [n]}$. In case of the generalized absolute value map $f(\lambda) = \lambda^{ n-1} \vert \lambda \vert, \lambda \in \mathbb{ R}$, we show that \[p^\ast p^{ n} \lesssim \Vert T_{ f^{ [n]}}: S_{ np} \times \ldots \times S_{ np} \rightarrow S_{ p} \Vert.\] This provides an alternative proof to one of the key theorems in the solution of Koplienko's problem on higher order spectral shift [Invent. Math. 193, No. 3, 501-538 (2013)], which is moreover sharp as $p \searrow 1$ and as $p \to\infty$ for any $n$.
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