The following remarkable inequality is due to the Hungarian mathematician P. Turn: If P n (x) denotes as usual Legendre's polynomial of the nth. degree, we have (1) A n (x) = (Pn(x)y -P w _i(*)P n+1 (*) 0, 1; -1 * S 1, with equality only f or x = 1. The purpose of this note is to give several proofs for this theorem different from that of Turn.
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Gábor Szegő (1948) studied this question.