A note on a theorem of Sunouchi

We show that for negativeα Sunouchi's formula $$begin{gathered} H_n (f,alpha ,beta ,x) = frac{1}{{A_n^beta }}sumnolimits_{k = 0}^n {A_{n - k}^{beta - 1} } |f(x) - sigma _k^alpha (f,x)|, hfill alpha > - frac{1}{2},beta > frac{1}{2}, hfill end{gathered}$$ becomes false, where σ _k ^α (f, x) is the (C,α) mean of the Fourier series for the functionf(x) ε Lipγ, 0<γ<1. A bound is given for H_n(f, α,β, x) for allα > -1,β> -1, which forα + β > 0, α≥ 0,β ≥0, coincides with the Sunouchi bound. The proof is by a method different from that of Sunouchi.