If $<math>f$</math> is Schur-convex and all first partial derivatives exist, then the following holds, where $<math> f_{(i)}(x)$ </math> denotes the partial derivative with respect to $<math> x_i$ </math>:
$$\end{equation} </math> for all $<math> x$ </math>. Since $<math> f$ </math> is a symmetric function., the above condition implies all the similar conditions for the remaining indexes!
== Examples ==
* $<math> f(x)=\min(x)$ </math> is Schur-concave while $<math> f(x)=\max(x)$ </math> is Schur-convex. This can be seen directly from the definition.
* The [[Shannon entropy]] function <math>\sum_{i=1}^d{P_i \cdot \log_2{\frac{1}{P_i}}}</math> is Schur-concave.
* The [[Rényi entropy]] funtion is also Schur-concave.
* <math>\ sum_{i=1}^d{x_i^k},k \ge 1 </math> is Schur-convex.
* The function $f(x) = \prod_{i=1}^n x_i is Schur-concave, when we assume all $x_i > 0$. In the same way, all the
