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We present a short proof of a conjecture proposed by I. Ra\c{s}a (2017),which is an inequality involving basic Bernstein polynomials and convexfunctions. This proof was given in the letter to I. Ra\c{s}a (2017). Themethods of our proof allow us to obtain some extended versions of thisinequality as well as other inequalities given by I. Ra\c{s}a. As a tool we usestochastic convex ordering relations. We propose also some generalizations ofthe binomial convex concentration inequality. We use it to insert someadditional expressions between left and right sides of the Ra\c{s}ainequalities.

A sharpening of a problem on Bernstein polynomials and convex function and related results