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Let $La(n,P)$ be the maximum size of a family of subsets of $[n]= \{1,2, ...,n \}$ not containing $P$ as a (weak) subposet, and let $h(P)$ be the length ofa longest chain in $P$. The best known upper bound for $La(n,P)$ in terms of$|P|$ and $h(P)$ is due to Chen and Li, who showed that $La(n,P) \le\frac{1}{m+1} \left(|P| + \frac{1}{2}(m^2 +3m-2)(h(P)-1) -1 \right) {\binom {n}{\lfloor n/2 \rfloor}}$ for any fixed $m \ge 1$.  In this paper we show that $La(n,P) \le \frac{1}{2^{k-1}} (|P| +(3k-5)2^{k-2}(h(P)-1) - 1 ) {n \choose {\lfloor n/2\rfloor} }$ for any fixed $k\ge 2$, improving the best known upper bound. By choosing $k$ appropriately, weobtain that $La(n,P) = O\left( h(P) \log_2\left(\frac{|P|}{h(P)}+2\right)\right) {n \choose \lfloor n/2 \rfloor }$ as a corollary, which we show is bestpossible for general $P$. We also give a different proof of this corollary byusing bounds for generalized diamonds. We also show that the Lubell function ofa family of subsets of $[n]$ not containing $P$ as an induced subposet is$O(n^c)$ for every $c>\frac{1}{2}$.

An improvement of the general bound on the largest family of subsets avoiding a subposet