<s>[INST] Mathematical Inequality Proof There are $4$ positive real numbers $\,a,b,c,d\,$ and one positive integer $\, p>0 $.
Prove:
$$
\left\lvert\sqrt[p]{a^p+b^p}-\sqrt[p]{c^p+d^p}\right\rvert^p
\leq \left\lvert a-c\right\rvert^p + \left\lvert b-d\right\rvert^p
$$
I have proved the cases $\,p=1$ and $\,p=2$. But the general case is difficult to prove.
Can anyone help me to finish the final prove for the general case? Thanks.
When $\,p=1,\,$
$LHS =\left\lvert (a+b)-(c+d)\right\rvert=\left\lvert (a-c)+(b-d)\right\rvert\leq \left\lvert a-c\right\rvert+\left\lvert b-d\right\rvert$=RHS.
When $\,p=2,\,$
$LHS =\left\lvert\sqrt{a^2+b^2}-\sqrt{c^2+d^2}\right\rvert^2=\left(a^2+b^2\right)-2\sqrt{a^2+b^2}\sqrt{c^2+d^2}+\left(c^2+d^2\right)\,$
$RHS =a^2-2ac+c^2+b^2-2bd+d^2\,$
$RHS-LHS \geq 0$
if and only if $\,-2ac-2bd \geq -2\sqrt{a^2+b^2}\sqrt{c^2+d^2}$
if and only if $\,\big(ac+bd\big)^2\leq \big(a^2+b^2\big)\big(c^2+d^2\big)\,$
if and only if $\,2acbd\leq a^2d^2+b^2c^2\,$
if and only if $\,\big(ad-bc\big)^2\geq 0\,$ which is true and $\,RHS \geq LHS$
 [/INST] The inequality is equivalent to
$$\left|(a^q+b^q)^{1/q}-(c^q+b^q)^{1/q}\right|\le \left(|a-c|^q+|b-d|^q\right)^{1/q}.$$
Up to a change of notation (|a-c|=x, |b-d|=y) and by symmetry, we only need to show
$$((x+a)^q+(y+b)^q)^{1/q}\le (x^q+y^q)^{1/q}+(a^q+b^q)^{1/q},$$
which is a simple case of the Minkowski Inequality. 
</s>