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A Banach space $X$ has \textit{property $(K)$}, whenever every weak* nullsequence in the dual space admits a convex block subsequence$(f_{n})_{n=1}^\infty$ so that $\langle f_{n},x_{n}\rangle\to 0$ as $n\to\infty$ for every weakly null sequence $(x_{n})_{n=1}^\infty$ in $X$; $X$ has\textit{property $(\mu^{s})$} if every weak$^{*}$ null sequence in $X^{*}$admits a subsequence so that all of its subsequences are Ces\`{a}ro convergentto $0$ with respect to the Mackey topology. Both property $(\mu^{s})$ andreflexivity (or even the Grothendieck property) imply property $(K)$. In thepresent paper we propose natural ways for quantifying the aforementionedproperties in the spirit of recent results concerning other familiar propertiesof Banach spaces.

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