Let $a_1, a_2, a_3$; $b_1, b_2, b_3$; $c_1, c_2, c_3$; $d_1, d_2, d_3$ be all real numbers.

We need to show that $$\begin{align}(a_1b_1c_1d_1 + a_2b_2c_2d_2 &+ a_3b_3c_3d_3)^4\\ &\leq (a_1^4+a_2^4+a_3^4)(b_1^4+b_2^4+b_3^4)(c_1^4+c_2^4+c_3^4)(d_1^4+d_2^4+d_3^4)\end{align}$$

I have used Cauchy-Schwarz inequality on $(a_1,a_2,a_3)$ and $(A_1,A_2,A_3)$ where $A_i=b_ic_id_i$ and get the following $$(a_1A_1+a_2A_2+a_3A_3)< (a_1^2+a_3^2+a_3^2)^{1/2}(A_1^2+A_3^2+A_3^2)^{1/2}$$ Next my aim was to assume $B_i=c_id_i$ and go forward. But this does not help and fails therefore.

  • $\begingroup$ Hi and welcome to the site! Since this is a site that encourages learning, you will get much more help if you show us what you have already done. Could you edit your question with your thoughts and ideas? $\endgroup$ – 5xum Jan 29 '15 at 12:43
  • $\begingroup$ A warm welcome from me too. You should add what you tried to the question post, rather than the comment section. Also, if your trying to respond specifically to @5xum , use the @ nameoftheuser, this way he gets notified you have replied. $\endgroup$ – The Artist Jan 29 '15 at 12:57
  • $\begingroup$ OK @ The Artist $\endgroup$ – user1942348 Jan 29 '15 at 13:00
  • $\begingroup$ @user1942348 , ummm dont keep a space between @ and the name of the user next time :) $\endgroup$ – The Artist Jan 29 '15 at 13:05
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    $\begingroup$ Not entirely sure why your post was voted closed, but here is a hint on how to use CS twice to prove that - $(a_1^4+a_2^4+a_3^4)(b_1^4+b_2^4+b_3^4)(c_1^4+c_2^4+c_3^4)(d_1^4+d_2^4+d_3^4) \ge (a_1^2b_1^2+...)^2(c_1^2d_1^2+...)^2 \ge (a_1b_1c_1d_1+...)^4$ $\endgroup$ – Macavity Jan 29 '15 at 13:35

Your inequality is an application of the Hölder's Inequality.

More specifically,

$$(a_1^4 + a_2^4 + a_3^4)^\frac{1}{4}\cdots(d_1^4 + d_2^4 + d_3^4)^\frac{1}{4} \ge \left[ a_1^{\left(4 \cdot \frac{1}{4}\right)}b_1^{\left(4 \cdot \frac{1}{4}\right)}c_1^{\left(4 \cdot \frac{1}{4}\right)}d_1^{\left(4 \cdot \frac{1}{4}\right)} + \cdots + a_4^{\left(4 \cdot \frac{1}{4}\right)}b_4^{\left(4 \cdot \frac{1}{4}\right)}c_4^{\left(4 \cdot \frac{1}{4}\right)}d_4^{\left(4 \cdot \frac{1}{4}\right)}\right]$$

$$\iff (a_1^4 + a_2^4 + a_3^4)\cdots(d_1^4 + d_2^4 + d_3^4) \ge (a_1b_1c_1d_1 + \cdots + a_4b_4c_4d_4)^4$$


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