In this paper, we prove the well-known Erdős–Lax inequality [4] in a sharpened form. As a consequence, another widely used inequality due to Ankeny and Rivlin [1] gets sharpened. These results may be useful in various applications that required the Erdős–Lax and the Ankeny–Rivlin inequalities.

In this paper, we are interested in obtaining large values of Dirichlet L-functions evaluated at zeros of a class of L-functions, that is,

$$ \begin{align*}\max_{\substack{F(\rho)=0\\ T\leq \Im \rho \leq 2T}}L(\rho,\chi), \end{align*} $$

$\chi $

$GL(n)$

${\mathbb {Q}}$

Let $p:\mathbb{C}\rightarrow \mathbb{C}$ be a polynomial. The Gauss–Lucas theorem states that its critical points, $p^{\prime }(z)=0$, are contained in the convex hull of its roots. We prove a stability version whose simplest form is as follows: suppose that $p$ has $n+m$ roots, where $n$ are inside the unit disk,

$$\begin{eqnarray}\max _{1\leq i\leq n}|a_{i}|\leq 1~\text{and}~m~\text{are outside}~\min _{n+1\leq i\leq n+m}|a_{i}|\geq d>1+\frac{2m}{n};\end{eqnarray}$$

$p^{\prime }$

$n-1$

$m$

$(dn-m)/(n+m)>1$

$n$

$m$

${\sim}n^{-1}$

We investigate the numbers of complex zeros of Littlewood polynomials $p\left( z \right)$ (polynomials with coefficients {−1, 1}) inside or on the unit circle $\left| z \right|\,=\,1$, denoted by $N\left( p \right)$ and $U\left( p \right)$, respectively. Two types of Littlewood polynomials are considered: Littlewood polynomials with one sign change in the sequence of coefficients and Littlewood polynomials with one negative coefficient. We obtain explicit formulas for $N\left( p \right)$, $U\left( p \right)$ for polynomials $p\left( z \right)$ of these types. We show that if $n\,+\,1$ is a prime number, then for each integer $k,\,0\,⩽\,k\,⩽\,n-1$, there exists a Littlewood polynomial $p\left( z \right)$ of degree $n$ with $N\left( p \right)\,=\,k$ and $U\left( p \right)\,=\,0$. Furthermore, we describe some cases where the ratios $N\left( p \right)/n$ and $U\left( p \right)/n$ have limits as $n\,\to \,\infty $ and find the corresponding limit values.

We study the sequence of monic polynomials orthogonal with respect to inner product

$$\begin{eqnarray*}\langle p, q\rangle = \int \nolimits \nolimits_{0}^{\infty } p(x)q(x){e}^{- x} {x}^{\alpha } \hspace{0.167em} dx+ Mp(\zeta )q(\zeta )+ N{p}^{\prime } (\zeta ){q}^{\prime } (\zeta ),\end{eqnarray*}$$

$\alpha \gt - 1$

$M\geq 0$

$N\geq 0$

$\zeta \lt 0$

$p$

$q$

S. Stahl conjectured that the zeros of genus polynomials are real. In this note, we disprove this conjecture.

We show that a system of r quadratic forms over a 𝔭-adic field in at least 4r+1 variables will have a non-trivial zero as soon as the cardinality of the residue field is large enough. In contrast, the Ax–Kochen theorem [J. Ax and S. Kochen, Diophantine problems over local fields. I, Amer. J. Math. 87 (1965), 605–630] requires the characteristic to be large in terms of the degree of the field over ℚp. The proofs use a 𝔭-adic minimization technique, together with counting arguments over the residue class field, based on considerations from algebraic geometry.

If μk,n(α,β) denotes the relative extrema of the Jacobi polynomial P(α,β)n(x), ordered so that μk+1,n(α,β) lies to the left of μk,n(α,β), then R. A. Askey has conjectured twenty years ago that for for k = 1,…, n — 1 and n = 1,2,=. In this paper, we give an asymptotic expansion for μk,n(α,β) when k is fixed and n → ∞, which corrects an earlier result of R. Cooper (1950). Furthermore, we show that Askey's conjecture is true at least in the asymptotic sense.

We consider the positive zeros j″vk, k = 1, 2,…, of the second derivative of the Bessel function Jν(x). We are interested first in how many zeros there are on the interval (0,jν1), where jν1 is the smallest positive zero of Jν(x). We show that there exists a number ƛ = —0.19937078… such that and . Moreover, j″v1 decreases to 0 and j″ν2 increases to j″01 as ν increases from ƛ to 0. Further, j″vk increases in —1 < ν< ∞, for k = 3,4,… Monotonicity properties are established also for ordinates, and the slopes at the ordinates, of the points of inflection when — 1 < ν < 0.