Let $a_1 \geq a_2 \geq \cdots \geq a_n \geq 0$ be a sequence of nonincreasing nonnegative real numbers. Define the set, for $t > 1$, $$ B_t = \Big\{b \in \mathbb{R}^n : b_i \geq 0, \sum_i b_i^2 \leq 1, \sum_i \sqrt{b_i} \leq t\Big\}. $$ I am interested in upper and lower bounds that differ only in constants (independent of $t, a$) for the following $$ f_t(a) = \max_{b \in B_t} \sum_{j=1}^n a_j b_j. $$ It can be seen as at the support function of the convex hull of $B_t$ for such nondecreasing nonnegative $a$.

Let $a_1 \geq a_2 \geq \cdots \geq a_n \geq 0$ be a sequence of nonincreasing nonnegative real numbers. Define the set, for $t > 1$, $$ B_t = \Big\{b \in \mathbb{R}^n : b_i \geq 0, \sum_i b_i^2 \leq 1, \sum_i \sqrt{b_i} \leq t\Big\}. $$ I am interested in upper and lower bounds that differ only in constants (independent of $t, a$) for the following $$ f_t(a) = \max_{b \in B_t} \sum_{j=1}^n a_j b_j. $$ It can be seen as at the support function of the convex hull of $B_t$ for such nonincreasing nonnegative $a$.