Given-r-t-j-gt-0-and-n-1-Prove-that-p-1-1-n-p-1-r-r-p-2-1-n-p-2-t-t-2-n-2-n-r-t-n-r-n-t-n-t-n-r-n-rt-p-2-1-n-p-2-t-t-p-3-1-n-p

Question Number 7507 by Master Moon last updated on 01/Sep/16

G i v e n r, t, j > 0 a n d n ⩾ 1; P r o v e t h a t

\frac{{[\underset{p_{1} = 1}{\sum^{n}} (p_{1}^{r} + r) + \underset{p_{2} = 1}{\sum^{n}} (p_{2}^{t} + t)]}^{2}}{n^{2} [{(n!)}^{\frac{r + t}{n}} + r {(n!)}^{\frac{t}{n}} + t {(n!)}^{\frac{r}{n}} + r t]} + \frac{{[\underset{p_{2} = 1}{\sum^{n}} (p_{2}^{t} + t) + \underset{p_{3} = 1}{\sum^{n}} (p_{3}^{j} + j)]}^{2}}{n^{2} [{(n!)}^{\frac{t + j}{n}} + t {(n!)}^{\frac{j}{n}} + j {(n!)}^{\frac{t}{n}} + t j]} ⩾ 8

B y : M r . C h h e a n g C h a n t r i a