For n ≥ 2, 1 ≤ k ≤ n-1 and positive reals x₁,…,xₙ with cyclic indices, let S_(n,k)(x)=Σᵢ₌₁ⁿ xᵢ/(xᵢ₊₁+…+xᵢ₊ₖ); k=2 is Shapiro's cyclic sum. At the equal point the sum is n/k, and the Diananda inequality at (n,k) asserts S_(n,k) ≥ n/k. Sheremet (arXiv:2606.05…
On the q-quadratic (Askey–Wilson) lattice, Area (arXiv:2608.04802) introduced the affine Bernstein basis Bⁿₖ(x)=binom nk((1+x)/(2))ᵏ((1-x)/(2))ⁿ⁻ᵏ and asked whether it satisfies a first-order lattice ladder σ DₓBⁿₖ=τ 𝕊ₓBⁿₖ with degσ ≤ 2 and degτ ≤ 1, where…
For a set S ⊆ ℤ let Mᵘ_(ℤ)χ_S be its uncentered discrete Hardy–Littlewood maximal function and let f^((k)) denote the k-th forward difference. Liao, Madrid, Palsson and Weigt define C_(k,p) as the least constant with norm(Mᵘ_(ℤ)χ_S)^((k))_(lp p(ℤ)) ≤ C_(k,p…
Let πₙ be the monic polynomial of degree n orthogonal on [-c,c], 0<c ≤ 1, with respect to the Jacobi weight (1-x)^α(1+x)^β, -1<α<β. Gautschi conjectured that [πₙ(-c)/πₙ(c)]² ((1-c)/(1+c))^(β-α)<1 for every n ≥ 1. Botta, Castillo and Tertuliano da Silva (arX…
Fan and Kadir have refuted the statement — formulated explicitly by Malikiosis and attributed by him to Shi — that every translational tile in a finite abelian p-group is spectral, by exhibiting three explicit tiles without spectra: a 64-point set Γ in Zm4⁴…
Let Lₙ be the Laguerre polynomial normalised by Lₙ(0)=1, so that the Lₙ are orthonormal for e⁻ˣ dx on (0,∞), and let Q(a,b,c,d)=∫₀^∞ Lₐ(x/2)L_b(x/2)L_c(x/2)L_d(x/2) e⁻ˣ dx, and let Q_S=[Q(a,S-a,c,S-c)]_(a,c=0)^(S). Gonçalves introduced Q_S to prove a sharpe…
In their study of filter integrals for orthogonal polynomials, Amdeberhan, Duncan, Moll and Sharma introduce the polynomials Xₙ(a) = 2²ⁿ⁻¹(a+1/2)ₙ-C(2n-1, n-1)(a)ₙ, prove that all n zeros of Xₙ are real and negative — their argument placing one at a=-n and…
