Let z n k = e i t n k {z_{nk}} = {e^{i{t_{nk}}}} , 0 ≤ t n 0 > ⋯ > t n n > 2 π 0 \leq {t_{n0}} > \cdots > {t_{nn}} > 2\pi , f f a function in the disc algebra A A , and μ n = max { | t n k − 2 k π / ( n + 1 ) | : 0 ≤ k ≤ n } {\mu _n} = \max \{ |{t_{nk}} - 2k\pi /(n + 1)|:0 \leq k \leq n\} . Denote by L n ( f ; ⋅ ) {L_n}(f;\; \cdot ) the polynomial of degree n n that agrees with f f at { z n k : k = 0 , … , n } \{ {z_{nk}}:k = 0, \ldots ,n\} . In this paper, we prove that for every p p , 0 > p > ∞ 0 > p > \infty , there exists a δ p > 0 {\delta _p} > 0 , such that | | L n ( f ; ⋅ ) − f | | p = O ( ω ( f ; 1 n ) ) ||{L_n}(f;\cdot ) - f|{|_p} = O(\omega (f;\frac {1} {n})) whenever μ n ≤ δ p / ( n + 1 ) {\mu _n} \leq {\delta _p}/(n + 1) . It must be emphasized that δ p {\delta _p} necessarily depends on p p , in the sense that there exists a family { z n k : k = 0 , … , n } \{ {z_{nk}}:k = 0, \ldots ,n\} with μ n = δ 2 / ( n + 1 ) {\mu _n} = {\delta _2}/(n + 1) and such that | | L n ( f ; ⋅ ) − f | | 2 = O ( ω ( f ; 1 n ) ) ||{L_n}(f;\cdot ) - f|{|_2} = O(\omega (f;\frac {1} {n})) for all f ∈ A f \in A , but sup { | | L n ( f ; ⋅ ) | | p : f ∈ A , | | f | | ∞ = 1 } \sup \{ ||{L_n}(f;\cdot )|{|_p}:f \in A,||f|{|_\infty } = 1\} diverges for sufficiently large values of p p . In establishing our estimates, we also derive a Marcinkiewicz-Zygmund type inequality for { z n k } \{ {z_{nk}}\} .