Upper bound of a quadratic, given the upper

wadiad6qxb

wadiad6qxb

Answered question

2022-04-10

Upper bound of a quadratic, given the upper bound of variable
Let p(x) be a quadratic polynomial p(x)=ax2+bx+c such that |p(x)|1 for |x|1. Prove that |cx2+bx+a|2 for |x|1.

Answer & Explanation

embemiaEffoset4rs

embemiaEffoset4rs

Beginner2022-04-11Added 14 answers

Employing a rather common strategy for such problems, from p(x)=ax2+bx+c, we get c=p(0),b=12p(1)p(1) and a=12[p(1)+p(1)]p(0).
Hence with q(x)=cx2+bx+a, we get
q(x)=p(0)x2+t12p(1)p(1)x+t12p(1)+p(1)p(0)
=p(0)(x21)+t12p(1)(x+1)t12p(1)(x1)

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