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Sub‐ and Superadditive Properties of Fejér's sine Polynomial
Author(s) -
Alzer Horst,
Koumandos Stamatis
Publication year - 2006
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609306018273
Subject(s) - mathematics , superadditivity , combinatorics , converse , sine , polynomial , mathematical analysis , geometry
LetS n ( x ) = ∑ k = 1 n( sin ( k x ) ) / kbe Fejér's sine polynomial. We prove the following statements.The inequality( S n ( x + y ) ) α( x + y ) rβ ⩽( S n ( x ) ) α x rβ +( S n ( y ) ) α y rβ ( n ∈ N ; α , rβ ∈ R )holds for all x , y ∈ (0, π) with x + y < π if and only if α ⩾ 0 and α + rβ ⩽ 1. The converse of the above inequality is valid for all x , y ∈ (0, π) with x + y < π if and only if α ⩽ 0 and α + rβ ⩾ 1. For all n ∈ N and x , y ∈ [0, π] we have 0 ⩽ S n ( x ) + S n ( y ) − S n ( x + y ) ⩽ 3 2 3 . Both bounds are best possible.2000 Mathematics Subject Classification 42A05, 26D05 (primary), 39B62 (secondary).