Result on the power of norm in Banach space?

Question

I want to inquire whether there exist any result of the type $$\|x+y\|^{\lambda} \leq c_1\|x\|^{\lambda}+ c_2\|y\|^{\lambda}$$ where $\lambda \in (0, 1]$, $c_1$ and $c_2$ are positive constants and $x, y$ are in banach space $X$? Any reference?

For $\lambda \geq 1$, i know that $$\|x+y\|^{\lambda} \leq 2^{\lambda-1}(\|x\|^{\lambda}+ \|y\|^{\lambda})$$

Answer 1

$||x+y||^{\lambda} \leq (2\max \{||x||,||y||\})^{\lambda}=2^{\lambda} \max \{||x||^{\lambda},||y||^{\lambda}\}) \leq 2^{\lambda} \{||x||^{\lambda}+||y||^{\lambda}\})$.