Can we generalize the triangle inequality for general $p >0?$
More precisely, for what values of $p >0,$ the triangle inequality $$|x+y|^p \leq |x|^p+|y|^p$$ holds for all $x,y \in \mathbb{R}$
If not do we have analogue of triangle inequality in some sense for $p>0?$
We have a strict inequality for $0 < p < 1$. For $p > 1$, the inequality can go either way depending on the signs of $x$ and $y$.
Proof. For positive $u, v \in \mathbb{R}$ and $q \geq 1$, we have
$$ (u + v)^q = u(u + v)^{q - 1} + v(u + v)^{q - 1} > u^q + v^q. $$
Pick some $x, y \in \mathbb{R}$ and $0 < p < 1$. Let $u = |x|^p$, $v = |y|^p$, and $q = 1 / p$. Then
$$ |x|^p + |y|^p = u + v > (u^q + v^q)^\frac{1}{q} = (|x| + |y|)^p \geq |x + y|^p. $$
This gives us the desired inequality $|x|^p + |y|^p > |x + y|^p$.