Inequalities for the Hurwitz zeta function

Author： Alzer Horst  

Publisher： Royal Society of Edinburgh

ISSN： 1473-7124

Source： Proceedings Section A: Mathematics - Royal Society of Edinburgh, Vol.130, Iss.6, 2000-12, pp. : 1227-1236

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Abstract

Let ζp(x) = ∑∞k=01/(x+k)p be the Hurwitz zeta function. Furthermore, let p > 1 and α ≠ 0 be real numbers and n > 2 be an integer. We determine the best possible constants a(p,α,n), A(p,α,n), b(p,n) and B(p,n) such that the inequalitiesa(p,α,n) < (ζp (∑nk=1xk))α/∑nk=1(ζp(xk))α < A(p,α,n)andb(p,n) < ζp(∏nk=1xk)/∏nk=1ζp(xk) < B(p,n)hold for all positive real numbers x1,...,xn.

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