The following theorem is proved.
{\bf Theorem.} {\it Let P(x)=βk=02nβakβxk be a polynomial with
positive coefficients. If the inequalities a2kβa2k+2βa2k+12ββ<cos2(n+2Οβ)1β hold for all k=0,1,...,nβ1, then
P(x)>0 for every xβR .}
We show that the constant cos2(n+2Οβ)1β in this theorem
could not be increased. We also present some corollaries of this theorem.Comment: Submitted to the journal "Mathematical Inequalities and Applications"
on September 29, 200