We show that if $a b c d = 1$, the sum $a + b + c + d$ cannot exceed a certain weighted mean of the expressions $\frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a}$ and $\frac{b}{a} + \frac{c}{b} + \frac{d}{c} + \frac{a}{d}$.
By applying the AM-GM inequality to the numbers $\frac{a}{b}$, $\frac{a}{b}$, $\frac{b}{c}$ and $\frac{a}{d}$, we obtain
$$a = \sqrt[4]{\frac{a^4}{a b c d}} = \sqrt[4]{\frac{a}{b} \cdot \frac{a}{b} \cdot \frac{b}{c} \cdot \frac{a}{d}} \leq \frac{1}{4}\left(\frac{a}{b} + \frac{a}{b} + \frac{b}{c} + \frac{a}{d}\right)$$
Analogously,
$$b \leq \frac{1}{4}\left(\frac{b}{c} + \frac{b}{c} + \frac{c}{d} + \frac{b}{a}\right), \quad c \leq \frac{1}{4}\left(\frac{c}{d} + \frac{c}{d} + \frac{d}{a} + \frac{c}{b}\right) \quad \text{ and } \quad d \leq \frac{1}{4}\left(\frac{d}{a} + \frac{d}{a} + \frac{a}{b} + \frac{d}{c}\right).$$
Summing up these estimates yields
$$a + b + c + d \leq \frac{3}{4}\left(\frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a}\right) + \frac{1}{4}\left(\frac{b}{a} + \frac{c}{b} + \frac{d}{c} + \frac{a}{d}\right).$$
In particular, if $a + b + c + d > \frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a}$ then $a + b + c + d < \frac{b}{a} + \frac{c}{b} + \frac{d}{c} + \frac{a}{d}$.