New inequalities related to sums of $L^p$ functions in connection with Carbery's problems

Carbery (2006) proposed novel estimates for the $L^p$ norm of a sum of two nonnegative measurable functions. Subsequently, Carlen, Frank, Ivanisvili and Lieb (2018) provided stronger bounds, which Ivanisvili and Mooney (2020) further refined to achieve estimates that are, in a certain sense, optimal. Continuing this line of research, the present work establishes new upper and lower bounds for the range $p\in (1,\mathrm{\infty })$. Carbery also asked under what conditions on a sequence $(f_j)$ of nonnegative measurable functions the inequality $\sum \mathrm{\parallel }{f}_j{\mathrm{\parallel }}_p^p<\mathrm{\infty }$ implies that $\sum f_j\in L^p$. Ivanisvili and Mooney (2020) resolved this question for $p\in [1,2]$, and the present work proposes an answer for $p\in [2,\mathrm{\infty })$.