Cómo probar $\sum (a+b+c+d)\sum\frac{ab+ac+ad+bc+bd+cd}{a+b+c+d}\sum\frac{abc+abd+bcd+cda}{ab+ac+ad+bc+bd+cd}\leq \sum a\sum b\sum c\sum d$

Question

dejar que $a{i}>0,b{i}>0,c{i}>0,d{i}>0,i=1,2,\cdots,n $

mostrar que $$\sum{i=1}^{n}(a{i}+b{i}+c{i}+d{i})\sum{i=1}^{n}\dfrac{a{i}b{i}+b{i}c{i}+c{i}d{i}+d{i}a{i}+a{i}c{i}+b{i}d{i}}{a{i}+b{i}+c{i}+d{i}}\sum{i=1}^{n}\dfrac{a{i}b{i}c{i}+b{i}c{i}d{i}+c{i}d{i}a{i}+d{i}a{i}b{i}}{b{i}c{i}+c{i}a{i}+a{i}b{i}+d{i}a{i}+d{i}b{i}+d{i}c{i}}\sum{i=1}^{n}\dfrac{a{i}b{i}c{i}d{i}}{a{i}b{i}c{i}+b{i}c{i}d{i}+c{i}d{i}a{i}+d{i}a{i}b{i}}\le\sum{i=1}^{n}a{i}\sum{i=1}^{n}b{i}\sum{i=1}^{n}c{i}\sum{i=1}^{n}d{i}$$

Creo que esto es cierto, porque ayer he demostrado esto

$\sum a+b+c \sum \frac{ab+bc+ca}{a+b+c} \sum \frac{abc}{ab+bc+ca} \leq \sum a \sum b \sum c$

Gracias a todos.

Answer 1

$$\sum a_i \sum bi \geq \sum ( a_i +b_i ) \sum \frac{a_i b_i}{a_i+b_i}( Milne)$$
$$\sum c_i \sum di \geq \sum ( c_i +d_i ) \sum \frac{c_i d_i}{c_i+d_i}( Milne)$$ $$\sum a_i \sum bi \sum c_i \sum di \geq \sum ( a_i +b_i ) \sum \frac{a_i b_i}{a_i+b_i} \sum ( c_i +d_i ) \sum \frac{c_i d_i}{c_i+d_i}$$ $$= \sum ( a_i +b_i ) \sum ( c_i +d_i )\sum \frac{a_i b_i}{a_i+b_i} \sum \frac{c_i d_i}{c_i+d_i}$$ $$(Milne ) \geq \sum ( a_i +b_i + c_i+d_i )\sum \frac{(a_i+ b_i)(c_i+d_i)}{a_i+b_i+c_i+d_i} \sum(\frac{a_i b_i}{a_i+b_i}+\frac{c_i d_i}{c_i+d_i})\sum \frac{\frac{a_i b_i }{a_i+b_i}\frac{c_i d_i}{c_i+d_i}}{\frac{a_i b_i }{a_i+b_i}+\frac{c_i d_i}{c_i+d_i}}$$ $$\geq \sum ( a_i +b_i + c_i+d_i )\sum \frac{(a_i+ b_i)(c_i+d_i)}{a_i+b_i+c_i+d_i} \sum \frac{a_ib_i(c_i+d_i)+c_id_i(a_i+b_i)}{(a_i+b_i)(c_i+d_i)} \sum \frac{a_ib_ic_id_i}{a_ib_i(c_i+d_i)+c_id_i(a_i+b_i)}$$

Ref : Dan Sitaru de "Imad Zak Math Group" en facebook