Abstract <p>This paper examines how well periodic functions and their conjugate functions in the generalized Hölder class can be approximated using the generalized Abel–Poisson and generalized conjugate Abel–Poisson operators, respectively. The quality of the approximation is assessed using two moduli of continuity and evaluated in terms of a specific norm. These findings offer a clear understanding of the performance of these operators in approximating periodic and conjugate functions. In particular, the degree of approximation of periodic functions and their conjugate functions by the classical Abel–Poisson integral and the conjugate Abel–Poisson integral, respectively, is derived as a special case of the main results.</p> <p> <b>MSC2020 numbers:</b>26A16; 41A25; 42A50; 26A15.</p>