Under the steady-state condition, the spectrum of electrons is investigated by solving the continuity equation under the complex radiation of both the synchrotron and Compton processes. The resulted gamma-ray burst (GRB) spectrum is a broken power law in both the fast and slow cooling phases. On the basis of this electron spectrum, the spectral indices of the Band function in four different phases are presented. In the complex radiation frame, the detail investigation on physical parameters reveals that three models can answer the a similar to -1 problem, which are the synchrotron plus synchrotron self-Compton in the internal and the external shock models, and the synchrotron plus the external Compton processes in the external shock model. A possible marginal to fast cooling phase transition in GRB 080916C is discussed. The time-resolved spectra in different main pulses of GRB 100724B, GRB 100826A and GRB 130606B are investigated. We found that the flux is proportional to the peak energy in almost all main pulses. A significant (5 sigma) correlation for F-p similar to E-p is evident the first main pulse of GRB 100826A, and three marginally significant (3 sigma) correlations F-p similar to E-p are found in main pulses of GRB 100826A and GRB 130606B. The correlation between spectral index and E-p at 3 similar to 4 sigma level are observed in the first main pulse of GRB 100826A. Such correlations are possible explained in the complex radiation scenario.