Summary: Estimation of the mean of a Poisson distribution using Varian’s asymmetric Linex loss functions [H. R. Varian, in S. E. Fienberg and A. Zellner (eds.), Studies in Bayesian econometrics and statistics. In honor of Leonard J. Savage (1975; Zbl 0365.62114), 195-208] is considered. The admissibility of the linear function of the sample mean is studied. Necessary and sufficient conditions on the inadmissibility of the linear estimator are also given. In particular, the sample mean is inadmissible (even for one dimension) for a class of Linex loss functions.