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 \(p\in(1,\infty)\). Carbery also asked under what conditions on a sequence \((f_j)\) of nonnegative measurable functions the inequality \(\sum \|f_j\|_p^p < \infty\) 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 \(p\in[2,\infty)\).