On explicit bounds in schottky' theorem

Abstract

Let ρ(z) denote the Poincaté metric i - {0,1}. It is known that

where λ(τ) is the elliptic modular function. we define the function . We obtain the following inequalities.

where 0 ⩽ ∞ ⩽ 1. By using the above inequalities and the Poincaré metric. we get a new version of Schottky's theorem:

where b=e ^π–16.

and if a < 1, K <1,

if a < 1, K >1,

AMS No.:

• 30C50