Let $a$ and $b$ a nonengative real numbers and $p$ a positive integer ,I am looking for some inequality like
$$|a-b|^p\leq f(a,b)-g(a,b)$$
where $f$ and $g$ are a positive functions in this form $f(a,b)=|a|.|b|^q$ and the same for $g(a,b)=|b|.|a|^q$
in other terms i need to see the minus in the upper bound for $|a-b|^p$
Any suggestions or books that brings this type of inequality is Welcome Thank you
Unfortunately, if $f(a,b)=|a|.|b|^q$ and $g(a,b)=|b|.|a|^q$ then $g(a,b)=f(a,b)$, so if $f(a,b)\ne f(b,a)$ then one of values $f(a,b)-g(a,b)$ and $f(b,a)-g(b,a)$ is negative, so one of inequalities $|a-b|^p\le f(a,b)-g(a,b)$ an $|b-a|^p\le f(b,a)-g(b,a)$ fails.
On the other hand, if $a\ge b$ then $(a-b)^p+b^p\le a^p$ because we can place in $p$-dimensional cube with side $a$ non-overlapping $p$-dimensional cubes with sides $a-b$ and $b$.