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Sensitivity \cite{CD82,CDR86} and block sensitivity \cite{Nisan91} are twoimportant complexity measures of Boolean functions. A longstanding open problemin decision tree complexity, the "Sensitivity versus Block Sensitivity"question, proposed by Nisan and Szegedy \cite{Nisan94} in 1992, is whetherthese two complexity measures are polynomially related, i.e., whether$bs(f)=O(s(f)^{O(1)})$.  We prove an new upper bound on block sensitivity in terms of sensitivity:$bs(f) \leq 2^{s(f)-1} s(f)$. Previously, the best upper bound on blocksensitivity was $bs(f) \leq (\frac{e}{\sqrt{2\pi}}) e^{s(f)} \sqrt{s(f)}$ byKenyon and Kutin \cite{KK}. We also prove that if $\min\{s_0(f),s_1(f)\}$ is aconstant, then sensitivity and block sensitivity are linearly related, i.e.$bs(f)=O(s(f))$.

New upper bound on block sensitivity and certificate complexity in terms of sensitivity