Help needed in understanding the following trick in solving the optimization problem

Question

Suppose in an optimization problem we have the following constraints $$\sum_{i=1}^K\frac{1}{x_{i}}\leq T,$$ $$0<x_i<X$$ where $X,T$ are some real constants. In this case, is it ok to use the following relation (due to Jensen inequality) $$K\frac{K}{\sum_{i=1}^Kx_i}\leq \sum_{i=1}^K\frac{1}{x_i}$$ and use the fact that the above Jenson inequality becomes equal when all $x_i$'s are equal. Then, replace the original constraint $\sum_{i=1}^K\frac{1}{x_{i}}\leq T$ in the optimization problem with the new constraint $$K\frac{K}{\sum_{i=1}^Kx_i}\leq T$$ with obviously all $x_i$'s in this constraint being equal. How we can justify this trick? Any help in this regard will be much appreciated. Thanks in advance.

Answer 1

There is no trick, if $a\leq b$ and $b\leq c$ then $a\leq c$:

$$K\frac{K}{\sum_{i=1}^Kx_i}\leq \sum_{i=1}^K\frac{1}{x_i}\;\;\;\;{\rm and }\;\;\;\;\;\sum_{i=1}^K\frac{1}{x_{i}}\leq T,$$ so $$K\frac{K}{\sum_{i=1}^Kx_i}\leq T$$

But of course, this is now weaker constrain then starting one.