Inequalities for Cyclic Functions Original Research Article

The nth cyclic function is defined byϕn(z)=∑ν=0∞ znν(nν)! (z∈C, 2⩽n∈N). We prove that if k is an integer with 1⩽k⩽n−1, then(n−k)! ϕ(k)n(x)xn−kα<ϕn(x)<(n−k)! ϕ(k)n(x)xn−kβ holds for all positive real numbers x with the best possible constantsα=1 and β= 2n-k over n.