Companions of the inequalities of Fejér--Jackson and Young |
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Authors: | Horst Alzer |
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Affiliation: | (1) Morsbacher str. 10, D-51545 Waldbröl, Germany;(2) Department of Mathematics and Statistics, University of Cyprus, P.O.Box 20537, 1678 Nicosia, Cyprus |
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Abstract: | Summary Applications of some well-known theorems of Jackson and Young lead to the sharp inequalities -1<nk-1Σ(cos(kx)+sin(kx))/k (n ≥1; 1<x<π) and -1/2Si(π)<nk-1Σ(cos(kx)·sin(kx))/k (n ≥1; xЄR) We prove that the following counterpart is valid for all integers n ≥1 and real numbers xЄ (0, π): -3/2≤nk-1Σ(cos(kx)-sin(kx))/k where the sign of equality holds if and only if n =2 and x = π /2. |
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