  Motivated by the inequality $\|f+g\|_{2}^{2} \leq
\|f\|_{2}^{2}+2\|fg\|_{1}+\|g\|^{2}_{2}$, Carbery (2006) raised the question
what is the "right" analogue of this estimate in $L^{p}$ for $p \neq 2$.
Carlen, Frank, Ivanisvili and Lieb (2018) recently obtained an $L^{p}$ version
of this inequality by providing upper bounds for $\|f+g\|_{p}^{p}$ in terms of
the quantities $\|f\|_{p}^{p}, \|g\|_{p}^{p}$ and $\|fg\|_{p/2}^{p/2}$ when $p
\in(0,1] \cup [2,\infty)$, and lower bounds when $p \in (-\infty,0) \cup
(1,2)$, thereby proving (and improving) the suggested possible inequalities of
Carbery. We continue investigation in this direction by refining the estimates
of Carlen, Frank, Ivanisvili and Lieb. We obtain upper bounds for $\|f +
g\|_p^p$ also when $p \in (-\infty,0) \cup (1,2)$ and lower bounds when $p \in
(0,1] \cup [2,\infty)$. For $p \in [1,2]$ we extend our upper bounds to any
finite number of functions. In addition, we show that all our upper and lower
bounds of $\|f+g\|_{p}^{p}$ for $p \in \mathbb{R}$, $p\neq 0$, are the best
possible in terms of the quantities $\|f\|_{p}^{p}, \|g\|_{p}^{p}$ and
$\|fg\|_{p/2}^{p/2}$, and we characterize the equality cases.
