Sharp inequalities for polygamma functions

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The main aim of this paper is to prove that the double inequality (k-1)!{x+[(k-1)!|ψ⁽ᵏ⁾(1)|]¹/ᵏ}ᵏ +k!xᵏ⁺¹<|ψ⁽ᵏ⁾(x)|<(k-1)!/(x+12)ᵏ+k!xᵏ⁺¹ holds for x>0 and k and that the constants [(k-1)!|ψ⁽ᵏ⁾(1)|]¹/ᵏ and 12 are the best possible. In passing, some related inequalities and (logarithmically) complete monotonicity results concerning the gamma, psi and polygamma functions are surveyed.