On the minimal degree and base size of finite primitive groups

Let $G$ be a finite permutation group acting on $\mathrm{\Omega }$. A base for $G$ is a subset $B\subseteq \mathrm{\Omega }$ such that the pointwise stabilizer $G_{(B)}$ is the identity. The base size of $G$, denoted by $b(G)$, is the cardinality of the smallest possible base. The minimal degree of $G$, denoted by $\mu (G)$, is the smallest cardinality of the support of a non trivial element of $G$. In this paper, we establish a new upper bound for $b(G)$ when $G$ is primitive, and subsequently prove that if $G$ is a primitive group different from the Mathieu group of degree $24$, then $\mu (G)b(G)\le n\log n$, where $n$ is the degree of $G$. This bound is best possible, up to a multiplicative constant.