Let K(α), 0≤α< 1, denote the class of functions g(z)= z+∑∞n=2anzn which are regular and univalently convex of order α in the unit disc U . Pursuing the problem initiated by Robinson in the present paper, among other things, we prove that if f is regular in U , f(0) = 0, and f(z)+zf ′(z) < g(z)+zg′(z) in U , then (i) f(z) < g(z) at least in |z| < r0, r0 = √ 5/3 = 0.745 . . . if f ∈ K; and (ii) f...