Examples of strongly asymmetric functions that are strictly Schur-convex

$$\def\R{\mathbb{R}}$$I have been looking for strongly asymmetric functions that are strictly Schur-convex, where the relevant concepts are defined below:

Definition 1: For function $$f: \R^n → \R$$ and permutation $$π$$ on $$\{1, 2, \cdots, n\}$$, if$$f(x_1, \cdots, x_n) = f(x_{π(1)}, \cdots, x_{π(n)}), \quad \forall x_1, \cdots, x_n \in \R$$ then $$f$$ is said to be $$π$$-symmetric. If $$f$$ is not $$π$$-symmetric for any non-identity permutation $$π$$, then $$f$$ is said to be strongly asymmetric.

Definition 2: Any function $$f: \R^n → \R$$ that satisfies the following condition is said to be strictly Schur-convex: For any $$x, y \in \R^n$$, if $$x_i > y_i \geqslant y_j > x_j$$, $$x_i + x_j = y_i + y_j$$ and $$x_k = y_k$$ ($$\forall k ≠ i, j$$) for some $$1 \leqslant i, j \leqslant n$$, then $$f(x) > f(y)$$.

Remark: The above definition of Schur-convexity is different from that in [1], where it requires the symmetry of $$f$$.

Current progress

For $$n = 2$$, I have found the following example:$$f_1(x_1, x_2) = |x_1 - x_2| + x_2.$$ To see that $$f_1$$ is strictly Schur-convex, define $$g_1(h; x) = f_1(x + h, x - h)$$ for $$x, h \in \R$$. Since$$g_1(h; x) = 2|h| + (x - h) = \begin{cases} x - 3h; & h \leqslant 0\\ x + h; & h > 0 \end{cases},$$ then $$g_1(h; x)$$ is strictly decreasing for $$h < 0$$ and strictly increasing for $$h \geqslant 0$$. Therefore $$f_1$$ is strictly Schur-convex.

Next, my intuition was to generalize this construction for $$n = 3$$ as$$f_2(x_1, x_2, x_3) = |x_1 - x_2| + |x_1 - x_3| + a_2 x_2 + a_3 x_3,$$ where $$a_2$$ and $$a_3$$ are distinct constants to be chosen. Analogously defining $$g_2(h; x, x_1) = f_2(x_1, x + h, x - h)$$ for $$x, x_1, h \in \R$$, then\begin{align*} g_2(h; x, x_1) &= |x_1 - x - h| + |x_1 - x + h| + (a_2 - a_3) h + (a_2 + a_3) x\\ &= \begin{cases} 2|x_1 - x| + (a_2 - a_3) h + (a_2 + a_3) x; & |h| \leqslant |x_1 - x|\\ 2|h| + (a_2 - a_3) h + (a_2 + a_3) x; & |h| > |x_1 - x| \end{cases}. \end{align*} Schur-convexity requires that $$g_2$$ be strictly decreasing for $$h \leqslant 0$$ and strictly increasing for $$h > 0$$, but considering $$g_2(h; x, x_1)$$ for $$|h| \leqslant |x_1 - x|$$ leads to a contradiction. So this might not be a correct way to find examples for $$n \geqslant 3$$.

Are there any asymmetric functions with explicit expressions that are strictly Schur-convex for $$n \geqslant 3$$? It would be better if such examples are sums of linear functions and/or their absolute values. Thanks in advance.

• What's the motivation for Definition 2? BTW, in the book, the symmetry is not required, it's a consequence of the properties of the order $\prec$ on $\mathbb{R}^n$. – user742759 Jun 23 at 16:24
• Also if you show your work how you found that example, that would save time for people trying to help you. – user742759 Jun 23 at 16:27
• @yumi Symmetry is implied by majorization but not by strict majorization, and definition 2 rewrites A.2.b in chapter 3, part I of the book. – Saad Jun 24 at 0:16
• @yumi The construction of $f_2$, sort of by chance, is by adding extra terms to $|x_1-x_2|$ to make $g_1$ increase and derease in respective regions. – Saad Jun 24 at 0:18