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The aim of this paper is to prove that Markov's theorem on variation of zeros of orthogonal polynomials on the real line (Markoff in Math Ann 27:177–182, 1886) remains essentially valid in the case of paraorthogonal polynomials on the unit circle. [ABSTRACT FROM AUTHOR]

Let P(z) be a polynomial of degree n which does not vanish in | z | < 1 . Then it was proved by Hans and Lal (Anal Math 40:105–115, 2014) that | z s P (s) + β n s 2 s P (z) | ≤ n s 2 (| 1 + β 2 s | + | β 2 s |) max | z | = 1 | P (z) | , for every β ∈ C with | β | ≤ 1 , 1 ≤ s ≤ n and | z | = 1. The L γ analog of the above inequality was recently given by Gulzar (Anal Math 42:339–352, 2016) who under the same hypothesis proved { ∫ 0 2 π | e i s θ P (s) (e i θ) + β n s 2 s P (e i θ) | γ d θ } 1 γ ≤ n s { ∫ 0 2 π | (1 + β 2 s ) e i α + β 2 s | γ d α } 1 γ { ∫ 0 2 π | P (e i θ) | γ d θ } 1 γ { ∫ 0 2 π | 1 + e i α | γ d α } 1 γ , where n s = n (n - 1) ... (n - s + 1) and 0 ≤ γ < ∞ . In this paper, we generalize this and some other related results. [ABSTRACT FROM AUTHOR]