A note on a theorem of Sunouchi

We show that for negative $\alpha$ Sunouchi's formula
\begin{gather*} H_n(f,\alpha,\beta,x)=\frac1{A^\beta_n}\sum_{k=0}^nA_{n-k}^{\beta-1}|f(x)-\sigma_k^\alpha(f,x)|,\\ \alpha>-\frac12,\quad\beta>\frac12, \end{gather*}
becomes false, where $\sigma_k^\alpha(f,x)$ is the $(C,\alpha)$ mean of the Fourier series for the function $f(x)\in\mathrm{Lip}\,\gamma$, $0<\gamma<1$. A bound is given for $H_n(f,\alpha,\beta,x)$ for all $\alpha>-1$, $\beta>-1$, which for $\alpha+\beta>0$, $\alpha\geqslant0$, $\beta\geqslant0$, coincides with the Sunouchi bound. The proof is by a method different from that of Sunouchi.