if $\max_i \frac{x_i}{c_i}>\max_i \frac{y_i}{c_i}$,whether $max_i\frac{x_i}{d_i}>\max_i \frac{y_i}{d_i}$ is right [closed]

Question

if $x_i,y_i,c_i,d_i>0$ all are monotonically decreasing sequences, $$\max_i \frac{x_i}{c_i}>\max_i \frac{y_i}{c_i}$$ then $$max_i\frac{x_i}{d_i} \geq \max_i \frac{y_i}{d_i}$$ can be derived？

Answer 1

Of course not. E.g., let

• $(x_1,x_2,x_3,,x_4,\dots)=(3,1,1,1\dots)$;
• $(y_1,y_2,y_3,y_4,\dots)=(2,2,2,2\dots)$;
• $(c_1,c_2,c_3,c_4,\dots)=(1,1,1,1,\dots)$;
• $(d_1,d_2,d_3,d_4,\dots)=(3,1,1,1,\dots)$.

Then $\max_i\dfrac{x_i}{c_i}=3>2=\max_i\dfrac{y_i}{c_i}$ but $\max_i\dfrac{x_i}{d_i}=1<2=\max_i\dfrac{y_i}{d_i}$.

Here each of the four sequences is decreasing only in the non-strict sense. If you insist on strict decrease, multiply the $i$th term of each of the four sequences by (say) $1/i$, thus keeping all the ratios $\dfrac{x_i}{c_i},\dfrac{y_i}{c_i},\dfrac{x_i}{d_i},\dfrac{y_i}{d_i}$ the same as before.