On a modified Hyers-Ulam sequence (Q1176162)

It is shown that, given a mapping \(f: E\to F\) between Banach spaces, which is continuous along rays through 0, and for which there is a \(\delta\geq 0\) and a \(0\leq p<1\) such that \[ \| f(x+y)-f(x)- f(y)\|\leq\delta\cdot(\| x\|^ p+\| y\|^ p)\text{ for all } x,y, \] there is a unique linear mapping \(T: E\to F\) such that for all integers \(k\geq 2\) one has \[ \| f(x)-T(x)\|\leq\delta\cdot\| x\|^ p\cdot {{k+\sum_{j=2}^{k-1}j^ k} \over {k-k^ p}} \text{ for all } x. \] In fact \(T(x)=\lim_{n\to\infty} {{f(k^ n x)}\over{k^ n}}\). The author poses the question whether the minimum is attained for \(k=2\).