Prove $x+y^\alpha z^{1-\alpha}\le (x+y)^\alpha(x+z)^{1-\alpha}$ for $x,y,z\ge 0$ and $0<\alpha<1$

Question

I wish to prove the inequality $$x+y^\alpha z^{1-\alpha}\le (x+y)^\alpha (x+z)^{1-\alpha}$$ for $x,y,z\ge 0$ real numbers, and $0<\alpha<1$. Clearly this holds when $x,y$ or $z$ are $0$, but I can't manage to prove this simply by differentiating. What other techniques are there that would allow me to prove this inequality?

Answer 1

With $u=(x^{\alpha},y^{\alpha})$ and $v=(x^{1-\alpha},z^{1-\alpha})$, then $u\cdot v\leq\|u\|_{1/\alpha}\|v\|_{1/(1-\alpha)}$. Simplifying this, the result follows.