Integral inequalities for self-reciprocal polynomials

Let n ≥ 1 be an integer and let P _n be the class of polynomials P of degree at most n satisfying z ⁿ P(1/z) = P(z) for all z ∈ C. Moreover, let r be an integer with 1 ≤ r ≤ n. Then we have for all P ∈ P _n :

$ \alpha _n (r)\int_0^{2\pi } {|P(e^{it} )|^2 dt} \leqslant \int_0^{2\pi } {|P^r (e^{it} )|^2 dt} \leqslant \beta _n (r)\int_0^{2\pi } {|P(e^{it} )|^2 dt} $ \alpha _n (r)\int_0^{2\pi } {|P(e^{it} )|^2 dt} \leqslant \int_0^{2\pi } {|P^r (e^{it} )|^2 dt} \leqslant \beta _n (r)\int_0^{2\pi } {|P(e^{it} )|^2 dt}
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