Inequalities for cyclic functions

The nth cyclic function is defined by
jn (z)=C
.
n=0
znn
(nn)! (z ¥ C, 2 [ n ¥ N).
We prove that if k is an integer with 1 [ k [ n−1, then
R (n − k)! j(k)
n (x)
xn−k S
a
< jn (x) < R (n − k)! j(k)
n (x)
xn−k S
b
holds for all positive real numbers x with the best possible constants
a=1 and b=R2n − k
n
S .