Let $z_{n,j}(\lambda)$ denote the positive zeros, in decreasing order, of the ultraspherical polynomial $C_n^\lambda$, $\lambda>-1/2$, with the reduced limiting interpretation at $\lambda=0$ specified below. Our principal result settles three higher-monotonicity questions of Gautschi: two as printed and the natural open-interval form of the third, whose printed endpoint $\lambda=0$ is singular. For every $n\geq3$, $$ \sqrt{\lambda+1}\,z_{n,j}(\lambda) $$ becomes a complete Bernstein function after translation of its parameter interval to $(0,\infty)$. For every $n\geq2$, $$ \sqrt{\lambda}\,z_{n,j}(\lambda) $$ is a complete Bernstein function on $(0,\infty)$. For every $n\geq4$, the largest-zero trajectory $$ \sqrt{\lambda+\frac{2n^2+1}{4n+2}}\,z_{n,1}(\lambda) $$ has the same property, whilst for every other positive zero the derivative of this scaling fails to be completely monotone. Separately, an exact calculation in degree $4$ provides a counterexample to a fourth conjecture of Gautschi, concerning the linear scaling $\lambda z_{n,j}(\lambda)$.
Let $R_n^{(a,b)}=P_n^{(a,b)}/P_n^{(a,b)}(1)$ be the normalised Jacobi polynomials. For $\alpha\ge0$, put $\xi_0=-1$ and let $\xi_n$ be the largest zero of $R_n^{(\alpha+1,\alpha)}$ for $n\ge1$. We prove that $$ R_k^{(\alpha,\alpha)}(x) =\min_{n\ge0}R_n^{(\alpha,\alpha)}(x), \quad \xi_{k-1}\le x\le\xi_k, $$ for every $k...
We study the zeros of the quasimodular forms $G_{\{2\}^n}$ defined by iterated sums. We first show that, for every $n>0$, $G_{\{2\}^n}$ has exactly $n$ simple zeros on each of the vertical half-lines $\Real(\tau)=0$ and $\Real(\tau)=1/2$, and that the zeros for consecutive values of $n$ satisfy an interlacing property....
Let $\varphi_{d}(|{\,\cdot\,}|)$ be the radial extremal function in the problem for the sharp Nikolskii constant $\mathcal C_d^{-1}=\inf \|f\|_{1}$ over functions $f\in\mathit{PW}\,_{1}^{1}(\mathbb R^{d})$, $f(0)=1$. For every odd dimension, we construct an entire function $\Phi$ of exponential type $1/2$ such that $\v...
Let $(\Omega,\mu)$ and $(\Lambda,\lambda)$ be complete atomless localizable semifinite measure spaces. Suppose that $E(\Omega,\mu)$ and $F(\Lambda,\lambda)$ are real rearrangement-invariant Banach function spaces with order-continuous norms, in the Banach-lattice sense, and that neither norm is proportional to the $L_2...
For positive integers $n\ge r$, let $\lambda(r,n)$ denote the \emph{maximal relative projection constant} of $r$-dimensional subspaces of $\ell_\infty^n$ and $\lambda(r)$ denote the \emph{maximal absolute projection constant}, respectively. It is known that for any fixed $r$, $\lambda(r,n)$ is a non-decreasing sequence...
Hitesh Kumar, Bojan Mohar, S. A. Mojallal et al.· 0 citations
We study the central Hardy--Littlewood maximal operator $M_c$ in $\mathbb{R}^n$ for all $n\geq1$, with particular emphasis on the question of its injectivity. More precisely, we consider whether there exist two nonnegative functions $f,g\in L^1(\mathbb{R}^n)$ such that $\|f-g\|_1>0$ and $M_cf=M_cg$. Along the way, we o...
A. Solyanik· 0 citations
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