On one extremal problem for numerical series - Radzievskaya, E. I.; Radzievskii, G. V. (Ukrains’kyi Matematychnyi Zhurnal, 2005)
Let $Γ$ be the set of all permutations of the natural series and let $α = \{α_j\}_{j ∈ ℕ},\; ν = \{ν_j\}_{j ∈ ℕ}$, and $η = {η_j}_{j ∈ ℕ}$ be nonnegative number sequences for which$$\left\| { u (α η )_γ } \right\|_1 : = \sum\limits_{j = 1}^∞ {v _j α _{γ (_j )} } η _{γ (_j )}$$is defined for all $γ:= \{γ(j)\}_{j ∈ ℕ} ∈ Γ$ and $η ∈ l_p$. We find $\sup _{η :\left\| η \right\|_p = 1} \inf _{γ \in Γ } \left\| { u (α η )_γ } \right\|_1$ in the case where $1 < p < ∞$.