This is a question which has puzzled me for a while. I would be very thankful if somebody can help me with it.
Assume you have $S$ rectangles appearing in front of your screen one by one. Each rectangle has the same area $A$ but not necessarily the same width and height. The first rectangle has width $w_{1}$ and height $h_{1}=A/w_{i}$, the second width $w_{2}$ and height $h_{2}=A/w_{2}$ etc. Then, the following inequality obviously holds:
$S\cdot \sum_{i=1}^{S}(w_{i}\cdot h_{i})\leq (\sum_{i=1}^{S}w_{i})\cdot(\sum_{j=1}^{S}h_{j})$. Note the left hand side is equal to $S^{2}\cdot A$.
My question is the following:
Assume, now, that the area is not necessarily the same, e.g you do not have a bijection between two elements of the two sets $W$ and $H$. You still have $S$ rectangles, the first with area $A_{1}\leq A$, the second with area $A_{2}\leq A$,... and the $S$-th with area $A_{S}\leq A$.
Assuming that the $i$-th rectangle has width $w_{i}$ and height $h_{i}$, the next one ($i+1$) has either:
height $h_{i+1}=h_{i}-1$ and width $w_{i+1}=w_{i}$ or
height $h_{i+1}\geq h_{i}$ and width $w_{i+1}=A/h_{i+1}$.
Can you say that $S\cdot(A_{1}+...+A_{S})\leq (\sum_{i=1}^{S}w_{i})\cdot(\sum_{j=1}^{S}h_{j})$ now?
Thank you.