Admissible solutions of the Schwarzian differential equation

Abstract

Abstract Let R ( z, w ) be a rational function of w with meromorphic coefficients. It is shown that if the Schwarzian equation possesses an admissible solution, then , where α j , are distinct complex constants. In particular, when R ( z, w ) is independent of z , it is shown that if (*) possesses an admissible solution w ( z ), then by some Möbius transformation u = ( aw + b ) / ( cw + d ) ( ad – bc ≠ 0), the equation can be reduced to one of the following forms: where τ j ( j = 1, … 4) are distinct constants, and σ j ( j = 1, … 4) are constants, not necessarily distinct.