Let $V$ be the Riemann sphere with unit diameter. Let $F_0$ be a connected domain on $V$, whose boundary $\partial F_0$ are analytic Jordan curves $\{\Lambda_i\}^q_{l = 1}$, where the spherical distance between any two curves is not smaller then $d \in (0,0.5)$. Let $L (\Lambda_i)$ be sets of all closed curves on $F_0$, or all curves on $F_0$, whose both end points lie on the same $\Lambda_i$. Consider the two domains which are surrounded by $\partial F_0$ and $c \in L (\Lambda_i)$, and let $D (\partial F_0, c) \subset F_0$ be one of the domains whose area is smaller. Let $$A (\Lambda_i) = \sup \left\{ {dD (\partial F_0, c) \over |c |}, c \in L (\Delta_i) \right\},$$ $$A(F_0) = \max \bigl\{ A (\Lambda_i); \ i = 1,2, \ldots, q \bigr\},$$ where $dD$ denotes the diameter of $D$, and $|c |$ denotes the measure of $c$. The $A (F_0)$ is called the cuted ratio of $F_0$. Let $F$ be a connected finite covering surface of $F_0$. Let $S = {|F |\over |F_0 |}$. We say that $S$ is the mean covering number of $F_0$. Let $L$ be the length of the relative boundary of $F$. We improve the Ahlforsâ€™s inequality as following: $$\rho^+ (F) > \rho (F_0) S - {32 \pi^2 A(F_0) L \over d^3},$$ where $\rho (F)$ is the characteristics of $F$, $\rho^+ = \max \{0, \rho\}$. If the boundary $\{\Lambda_i\}$ reduces to $q$ single points, or circles, then $\rho^+ (F) > \rho (F_0) S - {32 \pi^2 L \over d^3}$.

Reviewer: D.Sun