I am trying to understand the concept of weak majorization and majorization. Following the definitions in the book of Marshall et. al I have formulated the following reasoning. I suppose I am wrong somewhere, or something in the definitions is not clear to me yet. Can you please help me out here?
I know that
(1) if $x\geq y$, for $x,y \in \mathbb{R}^n_+$, one has $y\prec^w x$ and $y\prec_w x$, that is $y$ is both weakly supermajorized and weakly submajorized by $x$;
(2) $y\prec^w x$ and $y\prec_w x$ if and only if $y \prec x$ (i.e. $y$ is majorized by $x$)
(3) if $f:\mathbb{R}^n_+\rightarrow \mathbb{R}_+$ is Schur concave, then $y \prec x$ implies $f(x)\leq f(y)$.
Now, suppose $f:\mathbb{R}^n_+\rightarrow \mathbb{R}_+$ is continuous, strictly increasing and Schur concave. Let $x,y \in \mathbb{R}^n_+$ be such that $x>y$ (that is $x_i\geq y_i$ for all $i=1,...,n$ with at least one strict inequality), then we must have $f(x)>f(y)$.
However, by (1), $y\prec^w x$ and $y\prec_w x$ and, by (2) $y\prec x$. Since $f$ is Schur concave, $y\prec x$ implies $f(x)\leq f(y)$, which contradicts the previous inequality.
I suspect the problem comes from (2) because $\prec$ is defined for vectors such that $\sum_i x_i=\sum_i y_i$, while the weak majorization does not require this restriction. (1) and (2) are respectively euqations (15a) and (14) page 13 of the book.
Actually the mistake lies within the first point you mentioned, and it is a typo in the book of Marshall et al. Their equation (15a) on page 13 should read:
One readily verifies this by plugging $x_i\leq y_i$ (hence $x_{[i]}\leq y_{[i]}$ and $x_{(i)}\leq y_{(i)}$) into the definition of $\prec_w$ and $\prec^w$, respectively.