Some Sharp Inequalities for Algebraic Polynomials Original Research Article

LetHnbe the set of all algebraic polynomials with real coefficients of degree at mostn(n+1∈N). ForQ∈Hn,α, β>−1 we set‖Q‖α, β≔∫1−1 (1−x)α (1+x)β Q2(x) dx1/2.Letn+1∈N,ϕ(n; α, β; γ, δ)≔sup{‖Q‖α, β: Q∈Hn, ‖Q‖γ, δ=1}.In this paper we find explicit expressions in terms ofnforϕ(n; α, β; α+1, β),ϕ(n; α+1, β; α, β) in cases |α|=|β|=12, and for