A problem of inequality 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. 


Your inequality is an application of the Hölder's Inequality.
More specifically,
$$(a_1^4 + a_2^4 + a_3^4)^\frac{1}{4}\cdots(d_1^4 + d_2^4 + d_3^4)^\frac{1}{4} \ge \left[ a_1^{\left(4 \cdot \frac{1}{4}\right)}b_1^{\left(4 \cdot \frac{1}{4}\right)}c_1^{\left(4 \cdot \frac{1}{4}\right)}d_1^{\left(4 \cdot \frac{1}{4}\right)} + \cdots + a_4^{\left(4 \cdot \frac{1}{4}\right)}b_4^{\left(4 \cdot \frac{1}{4}\right)}c_4^{\left(4 \cdot \frac{1}{4}\right)}d_4^{\left(4 \cdot \frac{1}{4}\right)}\right]$$
$$\iff (a_1^4 + a_2^4 + a_3^4)\cdots(d_1^4 + d_2^4 + d_3^4) \ge (a_1b_1c_1d_1 + \cdots + a_4b_4c_4d_4)^4$$
