  We study general class of small-field axion inflations which are the mixture
of polynomial and sinusoidal functions suggested by the natural and axion
monodromy inflations. The axion decay constants leading to the successful axion
inflations are severely constrained in order not to spoil the Big-Bang
nucleosynthesis and overproduce the isocurvature perturbation originating from
the QCD axion. We in turn find that the cosmologically favorable axion decay
constants are typically of order the grand unification scale or the string
scale which is consistent with the prediction of closed string axions.
