Let be the Riemann sphere with unit diameter. Let be a connected domain on , whose boundary are analytic Jordan curves , where the spherical distance between any two curves is not smaller then . Let be sets of all closed curves on , or all curves on , whose both end points lie on the same . Consider the two domains which are surrounded by and , and let be one of the domains whose area is smaller. Let
where denotes the diameter of , and denotes the measure of . The is called the cuted ratio of . Let be a connected finite covering surface of . Let . We say that is the mean covering number of . Let be the length of the relative boundary of . We improve the Ahlfors’s inequality as following:
where is the characteristics of , . If the boundary reduces to single points, or circles, then .