This paper is concerned with the envelope-constrained H-infinity. filtering problem for a class of discrete nonlinear stochastic systems subject to quantization effects over a finite horizon. The system under investigation involves both deterministic and stochastic nonlinearities. The stochastic nonlinearity described by statistical means is quite general that includes several well-studied nonlinearities as its special cases. The output measurements are quantized by a logarithmic quantizer. Two performance indices, namely, the finite-horizon H-infinity specification and the envelope constraint criterion, are proposed to quantify the transient dynamics of the filtering errors over the specified time interval. The aim of the proposed problem is to construct a filter such that both the prespecified H-infinity, requirement and the envelope constraint are guaranteed simultaneously over a finite horizon. By resorting to the recursive matrix inequality approach, sufficient conditions are established for the existence of the desired filters. A numerical example is finally proposed to demonstrate the effectiveness of the developed filtering scheme. (C) 2018 Elsevier Ltd. All rights reserved.