Sub- And Superadditive Properties of Fejér's Sine Polynomial

Abstract

Let Sn (x) = n k =1 (sin(kx))/k be Fejér's sine polynomial. We prove the following statements. (i) The inequality (Sn (x + y)) α (x + y) β (Sn (x)) α x β + (Sn (y)) α y β (n ∈ N; α, β ∈ R) holds for all x, y ∈ (0, π) with x + y < π if and only if α 0 and α + β 1.(ii) 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 + β 1.(iii) For all n ∈ N and x, y ∈ [0, π] we have 0 Sn (x) + Sn (y) − Sn (x + y) 3 2 √ 3. Both bounds are best possible.

Some monotonic trigonometric sums

“…We determine all real numbers a, b, α, and β such that (1.4) and (1.5) hold for all n ≥ 1 and x ∈ (0, π). In [7] the authors made use of (1.3) in order to prove sub-and super-additive properties of x −→ (S n (x)) α x β (α, β ∈ R). Monotonic trigonometric sums have found some interesting applications in complex function theory and in particular in the theory of Bloch functions; see [8], [15].…”

“…Historical comments, various applications, as well as numerous references on this subject can be found in [1][2][3][4][5][6][7], [9], [11], and [17,Ch. 4].…”

The additivity of polygamma functions

“…Recall [3,5,7] that a function f is said to be sub-additive on I if f (x + y) ≤ f (x) + f (y) (1) holds for all x, y ∈ I such that x + y ∈ I. If the inequality (1) is reversed, then f is called super-additive on I.…”

“…A lot of literature for the sub-additive and super-additive functions can be found in [3,5] and related references therein.…”

Sub- and superadditive properties of Euler's gamma function

“…Sub-and superadditive functions have important applications in various fields, like functional analysis and semi-group theory [16], theory of differential equations [18], theory of convex sets [22], and statistics [28]. Moreover, sub-and superadditive problems are discussed in the theory of functional inequalities [12], [27], in number theory [14], [25], in the theory of trigonometric polynomials [3], and also in the theory of special functions [2], [4], [28].…”