caa a22 4500 388108444 CHVBK 20180307125346.0 cr unu---uuuuu 161130s1999 xx s 000 0 eng 10.1017/S0308210500021326 doi S0308210500021326 pii (NATIONALLICENCE)cambridge-10.1017/S0308210500021326 Alzer Horst Morsbacher Strasse 10, 51545 Waldbröl, Germany Some inequalities for arithmetic and geometric means [Elektronische Daten] [Horst Alzer] Let An and Gn (respectively, A′n and G′n) be the weighted arithmetic and geometric means of x1, ..., xn (respectively, 1 - x1, ..., 1 - xn). We present sharp upper and lower bounds for the differences and . And we determine the best possible constants r and s such that holds for all xi ∈ [a, b] (i = 1, ..., n; 0 < a < b < 1). Our theorems extend and sharpen results of Fan, Cartwright and Field, McGregor and the author. Copyright © Royal Society of Edinburgh 1999 Proceedings of the Royal Society of Edinburgh: Section A Mathematics Royal Society of Edinburgh Scotland Foundation 129/2(1999), 221-228 0308-2105 129:2<221 1999 129 PRM https://doi.org/10.1017/S0308210500021326 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1017/S0308210500021326 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Alzer Horst Morsbacher Strasse 10, 51545 Waldbröl, Germany NATIONALLICENCE 773 0- Proceedings of the Royal Society of Edinburgh: Section A Mathematics Royal Society of Edinburgh Scotland Foundation 129/2(1999), 221-228 0308-2105 129:2<221 1999 129 PRM CC0 http://creativecommons.org/publicdomain/zero/1.0 nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-cambridge