Let $G$ be a finite nonabelian simple group and $t$ is equal to the number of involutions of $G$. We know that $t<G/3$ or $3t+1 \leq G$. Is this the best upper bound for the number of involutions of $G$? and if not what is it (if there is any)?
If $n$ denotes the number of involutions of the nonAbelian simple group $G,$ then we have $n + 1 = \sum_{\chi} \nu(\chi)\chi(1),$ where $\chi$ runs over the irreducible characters of $G$ and $\nu(\chi) \in \{0,1,1\}$ denotes the FrobeniusSchur indicator of $\chi.$ Hence, by CauchySchwarz, we have $n < \sqrt{k(G)} \sqrt{G},$ where $k(G)$ denotes the number of conjugacy classes of $G.$ By a Theorem of Fulman and Guralnick, which uses the classification of finite simple groups, we have $k(G) < G^{0.41},$ so that $n < G^{0.705}.$ This is certainly less than $\frac{G}{4}$ when $G > 256.$ The only nonAbelian simple groups of order less than $256$ are $A_{5}$ and ${\rm PSL}(2,7),$ which have respectively $\frac{G}{4}$ and $\frac{G}{8}$ involutions. Hence every finite nonAbelian simple group $G$ has at most $\frac{G}{4}$ involutions and $4$ can't be replaced by any larger constant. However, the upper bound $G^{0.705}$ for the number of involutions in $G$ is asymptotically much stronger ( and I suspect this is far from best possible). In fact, it is possible to prove (without using the classification of finite simple groups I believe that R. Brauer knew this fact) that for any $\varepsilon > 0,$ there are only finitely many finite simple groups $G$ which have more than $\varepsilon G$ involutions. A more careful analysis of the argument at the beginning shows that if $d$ is the smallest degree of a nontrivial complex irreducible character of $G,$ then $G$ has less than $\frac{G}{d}$ involutions. But by Jordan's theorem on linear groups, for any integer $d >1,$ only finitely many nonAbelian simple groups have a nontrivial complex irreducible character of degree $d$ or less. In particular only finitely many nonAbelian simple groups have a nontrivial irreducible character of degree at most $\varepsilon^{1},$ so only finitely many nonAbelian simple groups $G$ have more than $\varepsilon G$ involutions. 

