Regular fractional iteration of convex functions

The existence of a unique ${\displaystyle C^1}$ solution φ of equation (1) is proved under the condition that f: I → I is convex or concave and of class ${\displaystyle C^1}$ in I, 0 < f(x) < x in I*, and f'(x) > 0 in I. Here I = [0, a] or [0, a), 0 < a ≤ ∞, and I* = I\ {0}.