We say that a nonnegatively curved manifold $(M,g)$ has quarter
pinched flag curvature if for any two planes which intersect in a line the ratio of their sectional
curvature is bounded above by $4$. We show that these manifolds have nonnegative complex sectional curvature.
By combining with a theorem of Brendle and Schoen it follows that any
positively curved manifold with strictly quarter pinched flag curvature must be a space form.
This in turn generalizes a result of Andrews and Nguyen in dimension 4.
For odd dimensional manifolds we obtain results for the case that the flag
curvature is pinched with some constant below one quarter, one of which generalizes a
recent work of Petersen and Tao.