Inequality, how to know intuition behind it

Question

I was solving the following inequality For $a$, $b$, $c$ and $d$ being positive real numbers which goes as $$ \frac{a}{a+b} + \frac{b}{b+c} + \frac{c}{c+d} + \frac{d}{d+a} \leq \frac{a}{b+c} + \frac{b}{c+d} + \frac{c}{d+a} + \frac{d}{a+b} $$ Which I was successful able to do But I am not able to understand the intuition behind the inequality, and how someone even came to it. Can someone help me to intuitively understand it

Answer 1

By Cauchy-Schwarz inequality,

$$(\frac{a-b}{b+c}+\frac{b-c}{c+d}+\frac{c-d}{a+d}+\frac{d-a}{a+b})(b+c+c+d+a+d+a+b)\ge(a-b+b-c+c-d+d-a)^2=0$$

We need to prove the first bracket is positive, but since $a,b,c,d$ are positive, it's indeed positive.

Equality occurs iff $a-b=b-c=c-d=d-a$, i.e. $a=b=c=d.$

The intuition behind this is that $\sum_{cyc}\frac a{a+b}$ is relatively closer to $1$ than $\sum_{cyc}\frac d{a+b}$. Consider this: the sum of the numerator and the denominator are the same, but RHS is more spread out. Then which side's is bigger? This is not mathematically rigorous, but that's my first instinct. Also, these inequalities can be made by permutating Cauchy-Schwarz, rearrangement inequality, and the Power Mean Inequality.