Gauss established a theory on the functional equation (Gauss' functional equation) 0} \right) \]" display="block"> f ( a + b 2 , a b ) = f ( a , b )   ( a , b > 0 ) , where f : R + × R + → R is an unknown function of the above equation.

In this paper we treat the functional equation 0} \right) \]" display="block"> f ( a + b 2 , 2 a b a + b ) = f ( a , b )   ( a , b > 0 ) where f : R + × R + → R is an unknown function of this equation.

The purpose of this paper is to prove new results on this functional equation by following the theory of Gauss' functional equation.