Let Ω">Ω be a bounded smooth domain in Rn">Rn, W1,n(Ω)">W1,n(Ω) be the Sobolev space on Ω">Ω, and λ(Ω)=inf{‖∇u‖nn:∫Ωudx=0,‖u‖n=1}">λ(Ω)=inf{‖∇u‖nn:∫Ωudx=0,‖u‖n=1} be the first nonzero Neumann eigenvalue of the n−">n−Laplace operator −Δn">−Δn on Ω">Ω. For 0≤α<λ(Ω)">0≤α<>, let us define ‖u‖1,αn=‖∇u‖nn−α‖u‖nn">‖u‖1,αn=‖∇u‖nn−α‖u‖nn. We prove, in this paper, the following improved Moser–Trudinger inequality on functions with mean value zero on Ω">Ω,supu∈W1,n(Ω),∫Ωudx=0,‖u‖1,α=1∫Ωeβn|u|nn−1dx<∞,">supu∈W1,n(Ω),∫Ωudx=0,‖u‖1,α=1∫Ωeβn|u|nn−1dx<∞,where βn=n(ωn−1∕2)1∕(n−1)">βn=n(ωn−1∕2)1∕(n−1), and ωn−1">ωn−1 denotes the surface area of unit sphere in Rn">Rn. We also show that this supremum is attained by some function u∗∈W1,n(Ω)">u∗∈W1,n(Ω) such that ∫Ωu∗dx=0">∫Ωu∗dx=0 and ‖u∗‖1,α=1">‖u∗‖1,α=1. This generalizes a result of Ngo and Nguyen (0000) in dimension two and a result of Yang (2007) for α=0">α=0, and improves a result of Cianchi (2005).