New bounds for $A_{infty}$ weights

Two new constants $\tilde{G_1}(u)$ and $\tilde{A_{\infty}}(u)$ are studied for weights $u: \mathbb R^n \rightarrow [0,\infty)$, which are simultaneously finite exactly for $A_{\infty}$ weights. The special case $v=h'$, $w=(h^{-1})'$ where $h : \mathbb R \rightarrow \mathbb R$ is an increasing homeomorphism induces the identity $\tilde{A_{\infty}}(w)=\tilde{G_1}(v)$. Other identities are established for such constants, when different measures are involved.