On sums of powers of inverse complete quotients

For an irrational number $x$, let $x_n$ denote its $n$-th continued fraction inverse complete quotient, obtained by deleting the first $n$ partial quotients. For any positive real number $r$, we establish the optimal linear bound on the sum of the $r$-th powers of the first $n$ complete quotients. That is, we find the smallest constants $\alpha(r), \beta(r)$ such that $x_1^r+\ldots +x_n^r< \alpha(r)n+\beta(r)$ for all $n\ge1$ and all irrationals $x$.