Quantifying properties ($K$) and (${\mu }^s$)

A Banach space $X$ has $\mathit{\text{property}}(K)$, whenever every weak* null sequence in the dual space admits a convex block subsequence $(f_n{)}_{n=1}^{\mathrm{\infty }}$ so that $⟨f_n,x_n⟩\to 0$ as $n\to \infty$foreveryweaklynullsequence$(x_{n})_{n=1}^\infty$in$X$;$X$ has $\mathit{\text{property}}({\mu }^s)$ if every weak${}^{\ast }$ null sequence in $X^{\ast }$ admits a subsequence so that all of its subsequences are Ces`{a}ro convergent to $0$ with respect to the Mackey topology. Both property $({\mu }^s)$ and reflexivity (or even the Grothendieck property) imply property $(K)$. In the present paper we propose natural ways for quantifying the aforementioned properties in the spirit of recent results concerning other familiar properties of Banach spaces.