We give a bound on the order of the Schur multiplier of $p$-groups refining earlier bounds. As an application we complete the classification of groups having Schur multiplier of maximum order. Finally we prove that the order of the Schur multiplier of a finite $p$-group of maximal class and order $p^n$ is at most $p^{\frac{n}{2}}$.
Classification of p-groups by their Schur multiplier
A classification of finite $p$-groups with a unique $\mathcal{A}_2$-subgroup
The classification of normalizing groups
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