If $\{Z_n\}$ is a Galton-Watson process with mean one, and $\tau$ is the extinction time, it is shown that $EZ^{1+\alpha} < \infty$ implies $E\tau^\beta = \infty$ for $\beta > 1/\alpha, 0 < \alpha < 1$. Conditions which imply $EZ^{1+\alpha} = \infty$ and $E\tau^\beta < \infty$ for $\beta < 1/\alpha, 0 < \alpha < 1$ are given. Necessary and sufficient conditions for $EX^{m+\alpha} < \infty$ or $EX^m \log X < \infty$ are given in terms of the Laplace transform of a general nonnegative random variable $X, 0 < \alpha < 1, m = 0, 1,\cdots$.