Question

Assume a = a(t) ∈ L1(0,1)∩C(0,∞), ∞ ≥ a(0+) > a(∞) ≥ 0 and a(t) ≥ 0, a′(t) ≤ 0, a′′(t) ≥ 0, a′′′(t) ≤ 0, on (0,∞). Let h(t) = ta(t) on (0,∞). Then one of the following is true. There exists a number ε > 0 such that h is increasing on (0, ε). Or, there is no such interval. Which is correct? Prove it.