I am trying to understand the properties of square variation, namely, the possibility of preserving it under certain operations. I am following Albiac & Kalton's book: Let $J$ stand for the usual definition of the James space (see Definition 3.4.1 p. 62).

Let $\mathcal{P}$ be the family of all non-increasing sequences of non-negative real numbers, convergent to 0.

Suppose that we are given a real function $f$ on positive integers taking non-negative values only. Assume, moreover, that:

$f(0)=0$

$(f(x_k))_{k=1}^\infty \in J\cap \mathcal{P}$

$\sum_{k=1}^n f(x_k+y_k)\leq \sum_{k=1}^n f(x_k)+\sum_{k=1}^n f(y_k)$

for every $(x_k)_{k=1}^\infty$ and $(y_k)_{k=1}^\infty$ in $c_0$.

Does the inequality hold $$\|(f(x_k+y_k))_{k=1}^\infty\|_J\leq \|(f(x_k))_{k=1}^\infty\|_J + \|(g(x_k))_{k=1}^\infty\|_J$$

In particular, must $(f(x_k+y_k))_{k=1}^\infty\in J$?

If the answer is negative, are there any sufficient conditions for $f$ to satisfy this inequality?