Inequalities for cyclic functions

The nth cyclic function is defined by φₙ(z)=∑ᵥ₌₀∞ zⁿᵛ/(nν)! (z∈ℂ, 2≤n∈ℕ). We prove that if k is an integer with 1≤k≤n−1, then (n−k)! φₙ^(k)(x)xⁿ⁻ᵏ⁻ᵅ < φₙ(x) < (n−k)! φₙ^(k)(x)xⁿ⁻ᵏ⁻ᵝ holds for all positive real numbers x with the best possible constants α=1 and β=(2n−k)/n.