Transcendental entire solutions of quadratic functional equations

Abstract

Solving Fermat-type functional equations is long standing problems in function theory and it is well-known that ƒ2 + g2 = 1 and ƒ2 + 2αƒg + g2 = 1 are respectively called binomial and trinomial quadratic equations. Using Nevanlinna theory as a tool, in this paper, we study existence and precise form of the solutions general quadratic functional equations aƒ2+2αƒg+bg2+2βƒ+2γg+C = 0. Moreover, we give an answer to a conjecture posed by Zhang. In addition, we obtain a result on the existence of meromorphic solutions to Fermat-type difference equations considering polynomial coefficients, which generalizes an existence result. Mathematics Subject Classification (2010). 39B32, 30D35.