Please present a sharp upper bound or a tight upper bound for $a^{1/q}-b^{1/q}$ ($a\geq b\geq 0$ and $q>1$).

Question

Please present a sharp upper bound or a tight upper bound for $a^{1/q}-b^{1/q},\;$ ($a\geq b\geq 0$ and $q>1$).

Answer 1

By Bernoulli's inequality, $$a^{1/q} - b^{1/q} = b^{1/q}\left(\left(1+\frac{a}b-1\right)^{1/q}-1 \right) \leqslant \frac{b^{1/q}}q\left(\frac{a}b-1 \right)$$ with equality when $a=b$.