Cauchy-Jensen functional inequality in Banach spaces and non-Archimedean Banach spaces (Q2912653)

Given fixed integers \(l\geq2, m\geq1, n\geq0\) and given a normed space \(X\) and a Banach space \(Y\) which also may be a non-Archimedean one, the authors investigate the stability of the inequality NEWLINE\[NEWLINE\left\|\sum_{i=1}^l f(x_i)+m\sum_{j=1}^n f(y_j)\right\|\leq\left\| m f(m^{-1} (\sum_{i=1}^l x_i)+\sum_{i=1}^n y_j)\right\|.NEWLINE\]NEWLINE They show that under certain conditions on \(\varphi\) the inequality NEWLINE\[NEWLINE\left\|\sum_{i=1}^l f(x_i)+m\sum_{j=1}^n f(y_j)\right\|\leq\left\| m f(m^-1 (\sum_{i=1}^l x_i)+\sum_{i=1}^n y_j)\right\|+\varphi(x_1,\dots,y_n)NEWLINE\]NEWLINE implies the existence of some function \(g\) satisfying the original inequality which is \textit{close} to \(f\) in some sense.NEWLINENEWLINEThe authors also claim that this \(g\) is additive. More precisely they say that it is easy to see that all solutions of the (first) inequality are additive. The assertion may be true. But certainly it cannot be seen easily. The special case \(l=2, m=1, n=0\) and \(Y=\mathbb{R}\) leads to the inequality \((f(x)+f(y))^2\leq f(x+y)^2\). The case of equality here gives a so-called \textit{alternative} equation. The solution of this equation [\textit{M. Kuczma}, An introduction to the theory of functional equations and inequalities. Basel: Birkhäuser (2009; Zbl 1221.39041), p. 380] is far from trivial. So the inequality case also should be rather involved.