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1

Cyclic mean-value inequalities for the gamma function

100%

Alzer H.

Colloquium Mathematicum

|

2013

|

tom 132

|

nr 1

27-34

EN

We present two cyclic inequalities involving the classical Γ-function of Euler and the (unweighted) power mean
$M_t(a,b) = ((a^t + b^t)/2)^{1/t}$ (t ≠ 0), M₀(a,b) = √(ab) (a,b>0).
(I) Let 2 ≤ n ∈ ℕ and r ∈ ℝ. The inequality
$∏_{j=1}^{n} Γ(1/(1 + M_r(x_j,x_{j+1}))) ≤ ∏_{j=1}^{n} Γ(1/(1 + x_j)) (x_{n+1} = x₁)$
holds for all $x_j > 0$ (j = 1,..., n) if and only if r ≤ 0. (II) Let 2 ≤ n ∈ ℕ and s ∈ ℝ. The inequality
$∏_{j=1}^{n} Γ(1/(1 + x_j)) ≤ ∏_{j=1}^{n} Γ(1/(1 + M_s(x_j,x_{j+1}))) (x_{n+1} = x₁)$
is valid for all $x_j > 0$ (j = 1,...,n) if and only if
$s ≥ max_{0 Here,
P(x) = 2x - 1 + x(x-1) ψ'(x)/ψ(x) and ψ = Γ'/Γ.

2

Inequalities for two sine polynomials

63%

Alzer H., Koumandos S.

Colloquium Mathematicum

|

2006

|

tom 105

|

nr 1

127-134

EN

We prove: (I) For all integers n ≥ 2 and real numbers x ∈ (0,π) we have $α ≤ ∑_{j=1}^{n-1} 1/(n²-j²) sin(jx) ≤ β$, with the best possible constant bounds α = (15-√2073)/10240 √(1998-10√2073) = -0.1171..., β = 1/3. (II) The inequality $0 < ∑_{j=1}^{n-1} (n²-j²)sin(jx)$ holds for all even integers n ≥ 2 and x ∈ (0,π), and also for all odd integers n ≥ 3 and x ∈ (0,π - π/n].

3

A sharp bound for a sine polynomial

63%

Alzer H., Koumandos S.

Colloquium Mathematicum

|

2003

|

tom 96

|

nr 1

83-91

EN

We prove that $|∑_{k=1}^{n} sin((2k-1)x)/k| < Si(π) = 1.8519...$ for all integers n ≥ 1 and real numbers x. The upper bound Si(π) is best possible. This result refines inequalities due to Fejér (1910) and Lenz (1951).

Ograniczanie wyników

3 Colloquium Mathematicum

3 Alzer H.

2 Koumandos S.

1 2013

1 2006

1 2003

Cyclic mean-value inequalities for the gamma function

Inequalities for two sine polynomials

A sharp bound for a sine polynomial