General hyperstability criteria for Jensen-type equations (Q6614070)

The author considers the classical Jensen functional equation \N\[\Nf\left(\frac{x+y}{2}\right)=\frac{f(x)+f(y)}{2} \tag{1}\N\]\Nand some of its modifications. Assuming that \(f\) takes its values in a normed space, the main query considered in the paper is to understand under which assumptions on the domain of \(f\) and on the {control function} \(\varphi\), every solution of the functional inequality \N\[\N\left\|f\left(\frac{x+y}{2}\right)-\frac{f(x)+f(y)}{2}\right\|\leq\varphi(x,y)\N\]\Nhas to be a solution of (1).