Fig 1.
Linear and piecewise-linear bounds.
The upper (blue) and lower (red) bounds and ν(k) > 0 (solid lines) are shown along with the true value (solid curve), the piecewise-linear bound that combines the recent linear bound for k ≥ 1 [3] with a chord segment (dashed lines), and the linear lower bound that is tangent at k = 1 (dash-dot line). The region k < 1 is not very usefully bounded.
Table 1.
Summary comparison of several upper and lower bounds. Gr&M refers to Groeneveld and Meeden 1977 [11], C&R refers to Chen and Rubin 1986 [4], B&P refers to Berg and Pedersen 2006 [6], and Ga&M refers to Gaunt and Merkle 2021 [3]. The * refers to bounds presented with informal proofs or derivations, and ** for bounds presented as conjectures without proof, in Lyon 2021 [1]; in the present paper they are treated as new theorems, with proofs.
Fig 2.
Upper (blue) and lower bounds (red) mapped for comparison to A(k) (black dotted). Bounds U6(k) and L8(k) define the top and bottom of the box via AU6(k) = AU = e−γ and AL8(k) = AL = log(2) − 1/3, while the left and right are defined by the limits of arctan(k) for 0 < k < ∞. To prove that the top and bottom are bounds of A(k), our approach is to find other upper and lower bounds “inside the box” over domains covering all k > 0. In this figure, we have no lower bound in the box around 1.7 < k < 3.0.
Fig 3.
Here we show some approximate-tangent lower bounds from Theorem L7 mapped to the box, and remove some of the others. The points {ki, νLi} of Theorem 7 are circled. The solid blue upper bound is in the box per Lemma 1 and Lemma 2. The solid red lower bound is in the box per Lemmas 3, 4, and 5. These bounds being in the box will prove our main theorems U6 and L8.
Table 2.
Lemmas to prove in support of Theorems U6 and L8.
Fig 4.
Conjectured better upper (blue) and lower (red) bounds mapped for comparison to A(k) (black dotted). Solid curves are from first-order rational-function interpolators; dashed curves are from arctan interpolators. For a sense of how close these bounds come to the median (50th percentile), curves of selected nearby percentiles are included (thin black curves, computed with Matlab’s gammaincinv function).