© 2015 Taylor & Francis. Purpose of this paper is to study some geometric properties of functions of a quaternion variable slice regular in the unit ball-like subordination, starlikeness, convexity and spirallikeness. In this framework, we prove some results, among which is the Rogosinski inequality. Moreover, we show some geometric and algebraic interpretations of our results in terms of maps from (Formula presented.) to itself. As a tool for subordination, we define a suitable notion of composition of slice regular functions which is of independent interest.