On combinations of the circle shifts and some one-dimensional integral operators

The diffeomorphism $\zeta=\zeta(e^{is})$ of the unit circle and the operator $\Psi \varphi(t) = \frac{1}{\pi i} \int\nolimits_{\Gamma} \left[\frac{\zeta'(\tau)}{\zeta(\tau)-\zeta(t)} - \frac{1}{\tau-t} \right] \varphi(\tau)d \tau$ are under consideration. The main results can be stated as follows: If $\zeta(t) \in C^{1,\alpha}(\Gamma)$, $0<\alpha\leqslant 1$, $\varphi(t) \in C^{0,\beta}(\Gamma)$, $0<\beta \leqslant 1$, $\mu=\alpha+\beta\leqslant 2$, then $\Psi \varphi (t) \in C^{\mu}(\Gamma)$ for $\mu < 1$. Moreover, the following inequality holds:
\begin{equation*} \|\Psi \varphi (t)\|_{C^{\mu}(\Gamma)} \leqslant {\rm const} \|\varphi(t)\|_{C^{0,\beta}(\Gamma)}, \end{equation*}
where the constant depends on $\|\zeta\|_{C^{1,\alpha}(\Gamma)}$ only. If $\mu=1$, then $ \Psi \varphi (t) \in C^{\mu -\varepsilon}(\Gamma)$ for all $0<\varepsilon<\mu$ and the similar inequality holds. If $\mu>1$, then $ \Psi \varphi (t) \in C^{1,\mu -1}(\Gamma)$, and
\begin{equation*} \|\Psi \varphi (t)\|_{C^{1,\mu-1}(\Gamma)} \leqslant {\rm const} \|\varphi(t)\|_{C^{0,\beta}(\Gamma)}, \end{equation*}
where the constant depends on $\|\zeta\|_{C^{1,\alpha}(\Gamma)}$ only. If $\zeta(t) \in C^{1,\alpha}(\Gamma)$, $0<\alpha\leqslant 1$, $\varphi(t) \in C^{1,\beta}(\Gamma)$, $0<\beta \leqslant 1$, then $ \Psi \varphi (t) \in C^{1,\alpha}(\Gamma)$, and
\begin{equation*} \|\Psi \varphi (t)\|_{C^{1,\alpha}(\Gamma)} \leqslant \mathrm{const}\, \|\varphi(t)\|_{C^{0,1}(\Gamma)} \leqslant \mathrm{const}\, \|\varphi(t)\|_{C^{1,\beta}(\Gamma)}, \end{equation*}
where the constant depends on $\|\zeta\|_{C^{1,\alpha}(\Gamma)}$ only. The index $\alpha$ in the left-hand side of the last inequality can not be improved. The appropriate example is given.