Superadditivity inequalities

Beckenbach, E. F.

doi:10.2140/pjm.1964.14.421

Cited by 16 publications

(8 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…, n) and 8 e (0, 1), then/ is said to be starshaped on R+. Interesting properties of these functions can be found in [6,7]. As an immediate consequence of the following theorem we obtain that the beta function is not starshaped on R+.…”

Section: =9+1mentioning

confidence: 58%

See 1 more Smart Citation

Inequalities for the beta function of n variables

Alzer

2003

ANZIAM J.

View full text Add to dashboard Cite

show abstract

Section: =9+1mentioning

confidence: 58%

“…. ,X n ) = = 0»(*»), (3)(4)(5)(6)(7)(8)(9) where 0 n is defined in (2.13). From Lemma 2.4 we get *» (*")>*" (c) = -l o g fl, (c), (3.10) so that (3.9) and (3.10) lead to 11) which is equivalent to the second inequality of (3.7).…”

Section: =9+1mentioning

confidence: 99%

Inequalities for the beta function of n variables

Alzer

2003

ANZIAM J.

View full text Add to dashboard Cite

show abstract

“…These functions have been studied in [8] and [9]. From (3.7) (with α = 1) we conclude that x → −S n (x) (n ∈ N) is star-shaped on [0, π].…”

Section: Remarks (1) Letmentioning

confidence: 93%

Sub- And Superadditive Properties of Fejér's Sine Polynomial

Alzer¹,

Koumandos²

2006

Bull. London Math. Soc.

View full text Add to dashboard Cite

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.

show abstract

“…Sub-and superadditive functions play an important role in various branches including, for example, probability, semi-groups, convex bodies, and differential equations. For more information on this subject we refer to [Bec64], [Bru60], [Bru62], [Bru64], [BO62], [HP57], [RS99], [Ros50], [TWW89].…”

Section: Introductionmentioning

confidence: 99%