How can I simplify -LSB- math -RSB- \ frac -LCB- 1 -RCB- -LCB- 1 + x ^ -LCB- m-n -RCB- -RCB- + \ frac -LCB- 1 -RCB- -LCB- 1 + x ^ -LCB- n-m -RCB- -RCB- ? -LSB- / math -RSB-
How can I prove that for all positive reals -LSB- math -RSB- x_1 , \ ldots , x_n -LSB- / math -RSB- with -LSB- math -RSB- x_1 x_2 \ ldots x_n = 1 -LSB- / math -RSB- we have -LSB- math -RSB- \ frac -LCB- 1 -RCB- -LCB- n-1 + x_1 -RCB- + \ ldots + \ frac -LCB- 1 -RCB- -LCB- n-1 + x_n -RCB- \ le 1 -LSB- / math -RSB- ?