Generalizations of Goncalves' inequality

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If F is a polynomial with complex coefficients, leading term aₙ, and roots α₁,..., αₙ, then Gonçalves' inequality states that F₂² is bounded below by aₙ² (∏ₙ₌₁ⁿ {1, αₙ²} + ∏ₙ₌₁ⁿ {1, αₙ²}). We establish generalizations of this inequality for other Lₚ norms, and derive additional lower bounds on the Lₚ norms of a polynomial in terms of its coefficients.