Let $\left\{a_k\right\}_{0\leqslant k\leqslant n}$ and $\left\{b_k\right\}_{0\leqslant k\leqslant n}$ be two positive decreasing sequences such that $c_1a_k\leqslant b_k\leqslant c_2 a_k$ for all $k$ for some positive constants $c_1,c_2$. Let $\epsilon>0$ and $f>0$ a function such that $$\sum_{k=0}^na_k\sin(k\epsilon)\geqslant f(n).$$

Question. Is it true that $$\sum_{k=0}^nb_k\sin(k\epsilon)\geqslant cf(n)$$ for some constant $c$ (independent of $n$, $\epsilon$)?

Let $\left\{a_k\right\}_{0\leqslant k\leqslant n}$ and $\left\{b_k\right\}_{0\leqslant k\leqslant n}$ be two positive decreasing sequences such that $c_1a_k\leqslant b_k\leqslant c_2 a_k$ for all $k$ for some positive constants $c_1,c_2$. Let $\epsilon>0$ and $f>0$ a function such that $$\sum_{k=0}^na_k\sin(k\epsilon)\geqslant f(n).$$

Question. Is it true that $$\sum_{k=0}^nb_k\sin(k\epsilon)\geqslant cf(n)$$ for some constant $c$ (independent of $n$, $\epsilon$)?

Without the monotonicity condition on the $a_k$'s and $b_k$'s, the answer is no, as shown in this answer.

Let $\left\{a_k\right\}_{0\leqslant k\leqslant n}$ and $\left\{b_k\right\}_{0\leqslant k\leqslant n}$ be two positive decreasing sequences such that $c_1a_k\leqslant b_k\leqslant c_2 a_k$ for all $k$ for some positive constants $c_1,c_2$. Let $\epsilon>0$ and $f$ a function such that $$\sum_{k=0}^na_k\sin(k\epsilon)\geqslant f(n).$$

Question. Is it true that $$\sum_{k=0}^nb_k\sin(k\epsilon)\geqslant cf(n)$$ for some constant $c$ (independent of $n$, $\epsilon$)?

Let $\left\{a_k\right\}_{0\leqslant k\leqslant n}$ and $\left\{b_k\right\}_{0\leqslant k\leqslant n}$ be two positive decreasing sequences such that $c_1a_k\leqslant b_k\leqslant c_2 a_k$ for all $k$ for some positive constants $c_1,c_2$. Let $\epsilon>0$ and $f>0$ a function such that $$\sum_{k=0}^na_k\sin(k\epsilon)\geqslant f(n).$$

Question. Is it true that $$\sum_{k=0}^nb_k\sin(k\epsilon)\geqslant cf(n)$$ for some constant $c$ (independent of $n$, $\epsilon$)?