caa a22 4500 605508194 CHVBK 20210128100635.0 cr unu---uuuuu 210128e20151201xx s 000 0 eng 10.1007/s00010-015-0371-1 doi (NATIONALLICENCE)springer-10.1007/s00010-015-0371-1 Alzer Horst Morsbacher Str. 10, 51545, Waldbröl, Germany aut The Hurwitz zeta function: monotonicity, convexity and inequalities [Elektronische Daten] [Horst Alzer] We present monotonicity and convexity properties of functions defined in terms of the Hurwitz zeta function $$\zeta(s, a)=\sum_{k=0}^\infty \frac{1}{(k+a)^s} \quad{(s > 1; \, a > 0)}$$ ζ ( s , a ) = ∑ k = 0 ∞ 1 ( k + a ) s ( s > 1 ; a > 0 ) and apply these results to obtain new inequalities involving $${\zeta(s, a)}$$ ζ ( s , a ) . Among others, we prove that the double-inequality $$\frac{1}{2} < a^s \zeta(s, a)+b^s \zeta(s, b)-(a+b)^s \zeta(s, a+b) < 1$$ 1 2 < a s ζ ( s , a ) + b s ζ ( s , b ) - ( a + b ) s ζ ( s , a + b ) < 1 holds for all s>1 and a, b > 0. Both bounds are sharp. Springer Basel, 2015 Hurwitz zeta function nationallicence monotonicity nationallicence convexity nationallicence inequalities nationallicence Euler numbers nationallicence Aequationes mathematicae Springer International Publishing 89/6(2015-12-01), 1401-1414 0001-9054 89:6<1401 2015 89 10 https://doi.org/10.1007/s00010-015-0371-1 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s00010-015-0371-1 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Alzer Horst Morsbacher Str. 10, 51545, Waldbröl, Germany aut NATIONALLICENCE 773 0- Aequationes mathematicae Springer International Publishing 89/6(2015-12-01), 1401-1414 0001-9054 89:6<1401 2015 89 10