ON LOWER BOUNDS FOR ERDOS-SZEKERES PRODUCTS

Let {s(j)}(j=1)(n) be positive integers. We show that for any 1 <= L <= n, parallel to Pi(n)(j=1)(1 - z(sj))parallel to(L infinity(vertical bar z vertical bar=1)) >= exp(1/2e L/(s(1)s(2) ... s(L))(1/L)). In particular, this gives geometric growth if a positive proportion of the {s(j)} are bounded. We also show that when the {s(j)} grow regularly and faster than j (log j)(2+epsilon), some epsilon > 0, then the norms grow faster than exp ((log n)(1+delta)) for some delta > 0.