Hall's Conjecture Research Articles

A proof of Hall's conjecture on length of ray images under starlike mappings of order α


 Assume that \(f\) lies in the class of starlike functions of order \(\alpha\in[0,1)\), that is, which are regular and univalent for \(|z|<1\) and such that
 Re\(\left(\frac{zf'(z)}{f(z)}\right)>\alpha\) for \(|z|<1.\)
 In this paper we show that for each \(\alpha\in[0,1)\), the following sharp inequality holds:
 \(|f(re^{i\theta})|^{-1}\int_{0}^{r}|f'(ue^{i\theta})|\,du\leq\frac{\Gamma(\frac{1}{2})\Gamma(2-\alpha)}{\Gamma(\frac{3}{2}-\alpha)}\) for every \(r<1\) and \(\theta\).
 This settles the conjecture of Hall (1980) positively.

Mean square of the derivatives of Hardy's Z-function

Let Z(t) be Hardy's Z-function. We shall study the asymptotic behavior of the derivatives of Z(t) with more precise error term. As an application of our formula, we shall give a proof of Hall's conjecture on the order of the error term in the mean square formula of Z(k)(t).

Hall’s Conjecture on Extremal Sets for Random Triangles

In this paper we partially resolve Hall's conjecture about the distribution of random triangles. We consider the probability that three points chosen uniformly at random, in a bounded convex region of the plane, form an acute triangle. Hall's conjecture is the "isoprobabilistic inequality" which states that this probability should be maximized by the disk. We first prove that the disk is a weak local maximum for planar regions and that the ball is a weak local maximum in three dimensional regions. We then prove a local $C^{2,1/2}$-type estimate on the probability in the Hausdorff topology. This enables us to prove that the disk is a local maximum in the Gromov-Hausdorff topology (modulo congruences). Finally, we give an explicit upper bound on the isoperimetric ratio of the regions which maximize the probability and show how this reduces verifying the full conjecture to a finite, though currently intractable, calculation. An interesting aspect of our work is the use of tools from outside of geometric probability. We use an autocorrelation integral to provide the appropriate framework for the problem. When we study the problem in $\mathbb{R}^3$, we need non-Abelian harmonic analysis. To set up the proof that the disk is a strong local maximum, we use the Borsak-Ulam theorem.

Search for good examples of Hall’s conjecture

Search for good examples of Hall’s conjecture

On Hall's conjecture

We show that for any even positive integer d there exist polynomials x and y with integer coefficients such that deg(x) = 2d, deg(y) = 3d and deg(x^3 - y^2) = d + 5.

On Mordell's equation

In an earlier paper we developed an algorithm for computing all integral points on elliptic curves over the rationals Q. Here we illustrate our method by applying it to Mordell's Equation y2=x3+k for 0 ≠ k ∈ Z and draw some conclusions from our numerical findings. In fact we solve Mordell's Equation in Z for all integers k within the range 0 < | k | ≤ 10 000 and partially extend the computations to 0 < | k | ≤ 100 000. For these values of k, the constant in Hall's conjecture turns out to be C=5. Some other interesting observations are made concerning large integer points, large generators of the Mordell–Weil group and large Tate–Shafarevic groups. Three graphs illustrate the distribution of integer points in dependence on the parameter k. One interesting feature is the occurrence of lines in the graphs.

A Proof of Hall's Conjecture on Starlike Mappings

A Proof of Hall's Conjecture on Starlike Mappings

On Wilbrink's Theorem

An easy extension of Wilbrink's Theorem on planar difference sets for higher values of λ is given. It follows that the Bruck-Ryser-Hall conjecture on the super-fluousness of the “ p > λ” condition (or its variations thereof) in the multiplier theorems implies Hall's conjecture on the existence of only a finite number of (ν, κ, λ) abelian difference sets, for the case λ odd and κ − λ ≡ 2 (mod 4). More strongly, under these conditions we can show that the corresponding designs are Hadamard, i.e., k = 2 λ + 1 and ν = 2 k + 1.

Diophantine equation x3?y2=k and Hall's conjecture

Diophantine equation x3?y2=k and Hall's conjecture

A lower bound on the permanent of a $(0,\,1)$-matrix

The paper gives a lower bound for the permanent of a fully indecomposable (0, 1)-matrix with at least k ones in each row. The result extends those of H. Minc and P. Gibson. Introduction. The computation of lower bounds for the permanent of a (0, 1)-matrix moved a step forward with the solution of Marshall Hall's conjecture by Richard Sinkhorn [5]. This was done by using, as a tool, the nearly decomposable matrix. Sinkhorn's argument rested heavily on the structure of this matrix which readily lends itself to inductive proofs. In [2] the structure of this matrix was simplified, thereby simplifying proofs for the above results. In [1] and [3] more efficient counting methods were used to improve lower bounds. The result of this paper further develops counting methods. These then are used to extend the results of Minc and Gibson. It is assumed that the reader is familiar with some of the language and notations of ([1], [2], [3], [4] and [5]). Results. All matrices discussed in this work are n x n (0, 1)-matrices with n>3. If A=(ai3) is a matrix then o(A)=, >j aij, and S=a(A)(3n-3). The following two lemmas are vital to our arguments. LEMMA 1. If A is nearly decomposable then A contains at most 3n-3 ones [4, p. 184]. LEMMA 2. If A has at least k>3 ones in each row then S>(k-3)n+3. PROOF. As a(A)?kn it follows that a(A)-(3n-3)>kn-(3n-3)= (k-3)n+3. Set R = S-(k-3)n. For k> 3 Lemma 2 implies R _ 3. Applying Lemma 1 it is seen that if A is fully indecomposable there are S ones in A which may be replaced by O's yielding A, with A still fully indecomposable. Applying Lemma 2 it is seen that (k-3)n of these ones may be replaced in A say n at a time yielding A1, A2,. , Ak-3 so that Ai (i= 1, 2, ... , k-3) Received by the editors April 21, 1972 and, in revised form, July 3, 1972. AMS (MOS) subject classifications (1970). Primary 15A15; Secondary 15A45, 15A48, 15A36. ? American Mathematical Society 1973