The Alexander polynomial of an alternating knot is written in the form \[ a_0- a_1(t+ t^{-1})+ a_2(t^2+ t^{-2})-\cdots+ (-1)^n a_n(t^n+ t^{-n}) \] such that \(a_i> 0\) for all \(i\) [K. Murasugi, Osaka J. Math. 10, 181-189 (1958; Zbl 0106.16606)]. In [J. Aust. Math. Soc., Ser. A 28, 241-249 (1979; Zbl 0442.57002)] R. I. Hartley proved that for a two-bridge knot, the coefficients satisfy \(a_0=\cdots= a_j> a_{j+1}>\cdots> a_n\;(>0)\) for a certain integer \(j\). The authors show that for a two-bridge knot, the leading coefficient \(a_n\) of Alexander polynomial imposes restrictions on all the other coefficients \(a_i\). More precisely, there exists a positive number \(c\) (depending on \(n\) and \(i\) not on the two-bridge knot) such that \(ca_n\geq a_i\). Moreover, they give an upper bound and a lower bound for \(a_{n- 1}\) in terms of \(a_n\). It is interesting that the leading coefficient restricts the other coefficients.