In this paper, we study the sequence \( (f(p_n))_{n \geq 1} \), where \( p_n \) is the nth prime number and \( f \) is a function of a class of slowly increasing functions including \( f(x) = \log_b x^r \) and \( f(x) = \log_b(\log_b x)^r \), where \( b \geq 2 \) is an integer and \( r > 0 \) is a real number. We give upper bounds of the discrepancy \( D^*_{N_i}(f(p_n), g) \) for a distribution function \( g \) and a sub-sequence \( (N_i)_{i \geq 1} \) of the natural numbers. Especially for \( f(x) = \log_b x^r \), we obtain the effective results for an upper bound of \( D^*_{N_i}(f(p_n), g) \).