Define $T^d$ as following $$ T^d = \left\{(t_1,\cdots,t_d)\in\mathbb{C}^{d}\mid |t_i|= 1 \mbox{ for all } i\right\} $$
For any partition $\lambda\vdash n$,The Schur function is defined
$$ \displaystyle s_\lambda(x_1, \ldots, x_d) = \frac{\det\Bigl(x_i^{d + \lambda_j -j}\Bigr)_{ij}}{\det\Bigl(x_i^{d-j}\Bigr)_{ij}}. $$
I would like to ask the (upper and lower) bounds in $u$ of $$ |s_\lambda(t)| $$ given that $|t_i-t_j|<u$ for all $i,j$.
I am also interested in bounding $|s_\lambda(t)|$ if there exists $i,j$ such that $|t_i-t_j|>u$