We consider the composition operator Cφ on the variable exponent Bloch space \(\cal{B}^{\alpha(\cdot)}\) , which consists of all analytic functions f on the unit disk \(\mathbb{D}\) such that \(\sup\{(1-{\mid z\mid}^{2})^{\alpha(z)}\mid f^{\prime}(z)\mid:z \in \mathbb{D}\}<\infty.\)
Here, α(z) is a log-Hölder continuous function on \(\mathbb{D}\) . The boundedness and compactness of Cφ are characterized. Besides, we show that \((1-{\mid z \mid}^{2})^{\alpha(z)} f^{\prime}(z)\) is Lipschitz continuous in terms of the pseudo-hyperbolic metric under the Lipschitz continuity of α(z). By using this result, we study the bounded and compact difference Cφ − Cφ of two composition operators on \(\cal{B}^{\alpha(\cdot)}\) , and the boundedness from below of Cφ is partially described.
