Tunable photonic elements at the surface of an optical fiber with piezoelectric core are proposed and analyzed theoretically. These elements are based on whispering gallery modes whose propagation along the fiber is fully controlled by nanoscale variation of the effective fiber radius, which can be tuned by means of a piezoelectric actuator embedded into the core. The developed theory allows one to express the introduced effective radius variation through the shape of the actuator and the voltage applied to it. In particular, the designs of a miniature tunable optical delay line and a miniature tunable dispersion compensator are presented. The potential application of the suggested model to the design of a miniature optical buffer is also discussed.