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We present a new, very short proof of a conjecture by I. Ra\c{s}a, which isan inequality involving basic Bernstein polynomials and convex functions. Itwas affirmed positively very recently by J. Mrowiec, T. Rajba and S.W\k{a}sowicz (2017) by the use of stochastic convex orders, as well as by Abel(2017) who simplified their proof. We give a useful sufficient condition forthe verification of some stochastic convex order relations, which in the caseof binomial distributions are equivalent to the I. Ra\c{s}a inequality. We givealso the corresponding inequalities for other distributions. Our methods allowus to give some extended versions of stochastic convex orderings as well as theI. Ra\c{s}a type inequalities. In particular, we prove the Muirhead typeinequality for convex orders for convolution polynomials of probabilitydistributions.

Muirhead inequality for convex orders and a problem of I. Ra\c{s}a on Bernstein polynomials