THE BOUNDARY PROPERTIES OF SOME LACUNARY TAYLOR-SERIES

Let the radius of convergence of the lacunary Taylor series f(z) = SIGMA-1infinity a(k) z(lambda-k) be 1. (1) If SIGMA \a(k)\ = infinity, under certain conditions, {f(e(it))} fills C when t varies in any arbitrarily small interval. (2) If SIGMA \a(k)\ = R is-an-element-of]0, + infinity[, under certain conditions, {z; Absolute value of z less-than-or-equal-to R} superset-of {f(z): Absolute value of z = 1} superset-of {z; Absolute value of less-than-or-equal-to R - delta-0}, where delta-0 is a positive constant.