For (4.18.1) see Copson (1935, p. 136).
(4.18.3) follows by the same method and (4.18.2) is
a consequence. (4.18.5) to (4.18.9) are
straightforward and (4.18.10) is obtained from the Maclaurin
expansions of and . For the second inequality in
(4.18.4), it is sufficient to show
for . The
function is zero at and and it has no zeros in
, because can have only
one zero in where intersects
. The first inequality is proved similarly.