Improved bounds for the disk-packing constant

```@article{Boyd1973ImprovedBF,
title={Improved bounds for the disk-packing constant},
author={D. W. Boyd},
journal={aequationes mathematicae},
year={1973},
volume={9},
pages={99-106}
}```
• D. W. Boyd
• Published 1973
• Mathematics
• aequationes mathematicae
This result lends considerable weight to the heuristic estimate S ~ 1.306951, obtained by Melzak [6]. The improvement is a result of some new inequalities involving the disk-packing function M(a, b, c; t). Our principal new result is that M is a strictly convex function of (a, b, c). This, combined with the fact that M is a symmetric function of these three variables and an auxiliary lemma, allows us to show that
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References

SHOWING 1-6 OF 6 REFERENCES
The disk-packing constant
The lower bound was subsequently improved by Wilker [8] to 1.059, and by the author [2] to 1.28467. An improved upper bound of 1.5403 . . . . (9+x/41)/10 was proved in [3], but the arguments there,Expand
Lower Bounds for the Disk Packing Constant
An osculatory packing of a disk, U, is an infinite sequence of disjoint disks, f Un }, contained in U, chosen so that, for n _ 2, U,, has the largest possible radius, r,,, of all disks fitting inExpand
On the Solid-Packing Constant for Circles
A solid packing of a circular disk U is a sequence of disjoint open circular subdisks Ul, U2, . whose total area equals that of U. The Mergelyan- Wesler theorem asserts that the sum of radiiExpand