Inequalities for cyclic functions

The nth cyclic function is defined by f(n)(z) = (infinity)Sigma (v=0) z(m)/(nv)! (z is an element of C, 2 less th an or equal to n is an element of N). We prove that if k an integer with 1 less than or equal to k less than or e qual to n-1, then ((n-k)!phi ((k))(n)(x)/x(n-k))alpha < phi (n)(x) < ((n-k)!phi ((k))(n)(x)/x (n-k))(beta) holds for all positive real numbers x with the best possible constants alpha = 1 and beta = ((2n-k)(n)). (C) 2001 Academic Press.