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This paper focuses on the performance analysis of a class of limited peak-to-average power ratio (PAPR) precoders for downlink multi-user massive multiple-input multiple-output (MIMO) systems. Contrary to conventional precoding approaches based on simple linear precoders such as maximum ratio transmission (MRT) and regularized zero-forcing (RZF), the precoders in this paper are obtained by solving a convex optimization problem. To be specific, these precoders are designed so that the power of each precoded symbol entry is restricted, and the PAPR at each antenna is tunable. By using the Convex Gaussian Min-max Theorem (CGMT), we analytically characterize the empirical distribution of the precoded vector and the joint empirical distribution between the distortion and the intended symbol vector. This allows us to study the performance of these precoders in terms of per-antenna power, per-user distortion power, signal-to-noise and distortion ratio (SINAD), and bit error probability. We show that for this class of precoders, there is an optimal transmit per-antenna power that maximizes the system performance in terms of SINAD and bit error probability.