A necessary correction: it seems that I was wrong in my calculations, not only in life

 The answer to the Problem in the post here is WRONG. It was brought to my attention by a commenter, unfortunately anonymous.

          The answer will be replaced there with this one :

Corrected   ANSWER CiP

The statement is NOT true for polynomials of degree 4 :

If  $P(x)=x^4\;,\;\;Q(x)=x^4+x^2\;,\;\; R(x)=2x^4+x^2\;\;$  then  $P(x)\leqslant Q(x)\leqslant R(x)$

but  $\lambda \cdot P(x)+(1-\lambda)\cdot R(x)=(2-\lambda)\cdot x^4+(1-\lambda)\cdot x^2\neq Q(x)$

$\square$