Convexity bound in Lieb and Loss. I'm currently reading through Lieb and Loss's Analysis text. At the end of the proof of theorem 1.9 the authors prove the inequality
$$ \left( |a|+|b|\right)^{p}\leq(1-\lambda)^{1-p}|a|^{p}+\lambda^{1-p}|b|^{p} $$
where $a,b\in \mathbb{C} , p>1$ and $ 0<\lambda<1$. They cite the convexity of the map $|t|^p$ on $0<t<1$. I'm having  trouble seeing this and I couldn't get the coefficients to work out so that I could use convexity. On the other hand, I managed to prove this estimate by setting
$$f(x)= \left(1-x \right)^{1-p}|a|^{p}+x^{1-p}|b|^{p}, 0<x<1$$ and using the second derivative to show that the minimum of $f$ occurs at $$\frac{|b|}{|b|+|a|}$$ and equals $$\left( |a|+|b|\right)^{p}.$$ I imagine a solution using the convexity of $|t|^p$ would be much cleaner.
I'd appreciate if anyone could illuminate this.

If $\lambda\in (0,1)$:
$$
(|a| + |b|)^p = \left[ (1-\lambda) \left(\frac{|a|}{1-\lambda}\right) +
\lambda \left(\frac{|b|}{\lambda}\right)\right]^p
\leq
(1-\lambda)\left(\frac{|a|}{1-\lambda}\right)^p
+\lambda \left(\frac{|b|}{\lambda}\right)^p.
$$
