quantiles with convex combination

Question

I have define the right quantile and left quantile, and I want to study the quantiles with convex combination. that's $$F^{-1}(q) = \inf\{x: F(x) \geq q\},$$ and $$F^{-1+}(q) = \inf\{x: F(x) > q\}.$$ is there some paper to study the following form quantile: $$F^{-1, \lambda}(q) = (1 - \lambda) F^{-1}(q) + \lambda F^{-1+}(q), \forall \lambda \in [0, 1].$$