On a relation between real and holomorphic functions

We study $\lambda$-holomorphic functions, which generalize holomorphic functions to the case of arbitrary complex exponent $\lambda$. We establish a connection between such functions and real-valued quadratic forms and prove that for $\lambda\ne\mu$, $\lambda\ne\overline{\mu}$ there are $\lambda$- and $\mu$-holomorphic functions whose imaginary parts coincide identically; such functions are polynomials of degree no greater tan two.