I'm trying to prove the following inequalities: $$ (1+a^2)^s \leq (1+(a-b)^2)^s + (1+b^2)^s$$ $$ 1 +a^2 \lesssim (1+(a-b)^2)(1+b^2).$$ They're in a set of notes I'm reading, just stated in passing. In application in the notes, the exponent $s \in [0,1/4)$, but I think it works for a broader range. The notation $x \lesssim y $ means $ x \leq Cy$, where $C$ is some absolute constant.
Both are clear if $|a| \leq |b|$ or if $|a| \leq |a-b|$. This leaves the case where $b$ lies between the origin and $a$.
I thought I could use the concavity of the function $x^s$ for the first one, but I haven't been able to make it work.
I think I found a way for the second. In the case where $b$ is between $0$ and $a/2$, we have $$ (1+(a-b)^2) \geq 1+a^2/4 \gtrsim 1+ a^2 \text{ since } 1+a^2 \leq 4(1+a^2/4). $$ When $b$ is between $a/2$ and $a$, we get $$ 1+ b^2 \geq 1+a^2/4 \gtrsim 1 +a^2.$$
However, this seems messy. Any ideas on how to prove the first, and in general how to approach inequalities like this? They keep popping up in my reading and I'd like to see how to prove them elegantly. Thanks.
