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We extend an inequality involving the Bernstein basis polynomials and convex functions on [0, 1]. The inequality was originally conjectured by Raşa about thirty years ago, but was proved only recently. Our extension provides an inequality involving  q -monotone functions,    $$q\in \mathbb N$$    q  ∈  N     . In particular, 1-monotone functions are nondecreasing functions, and 2-monotone functions are convex functions. In general,  q -monotone functions on [0, 1], for    $$q\ge 2$$    q  ≥  2     , possess a    $$(q-2)$$    (  q  -  2  )     nd derivative in (0, 1), which is convex there. We also discuss some other linear positive approximation processes.

An Extension of Raşa’s Conjecture to q-Monotone Functions