A sharp oscillation criterion for a difference equation with constant delay

It is known that all solutions of the difference equation $$\Delta x(n)+p(n)x(n-k)=0, \quad n\geq0, $$ Δ x ( n ) + p ( n ) x ( n − k ) = 0 , n ≥ 0 , where $\{p(n)\}_{n=0}^{\infty}$ { p ( n ) } n = 0 ∞ is a nonnegative sequence of reals and k is a natural number, oscillate if $\liminf_{n\rightarrow\infty}\sum_{i=n-k}^{n-1}p(i)> ( \frac {k}{k+1} ) ^{k+1}$ lim inf n → ∞ ∑ i = n − k n − 1 p ( i ) > ( k k + 1 ) k + 1 . In the case that $\sum_{i=n-k}^{n-1}p(i)$ ∑ i = n − k n − 1 p ( i ) is slowly varying at infinity, it is proved that the above result can be essentially improved by replacing the above condition with $\limsup_{n\rightarrow\infty}\sum_{i=n-k}^{n-1}p(i)> ( \frac{k}{k+1} ) ^{k+1}$ lim sup n → ∞ ∑ i = n − k n − 1 p ( i ) > ( k k + 1 ) k + 1 . An example illustrating the applicability and importance of the result is presented.

Growth estimates for the maximal term and central exponent of the derivative of a Dirichlet series

Let $A\in(-\infty,+\infty]$, $\Phi:[a,A)\to\mathbb{R}$ be a continuous function such that $x\sigma-\Phi(\sigma)\to-\infty$ as $\sigma\uparrow A$ for every $x\in\mathbb{R}$, $\widetilde{\Phi}(x)=\max\{x\sigma -\Phi(\sigma):\sigma\in [a,A)\}$ be the Young-conjugate function of $\Phi$, $\overline{\Phi}(x)=\widetilde{\Phi}(x)/x$ and $\Gamma(x)=(\widetilde{\Phi}(x)-\ln x)/x$ for all sufficiently large $x$, $(\lambda_n)$ be a nonnegative sequence increasing to $+\infty$, and $F(s)=\sum\limits\limits_{n=0}^\infty a_ne^{s\lambda_n}$ be a Dirichlet series such that its maximal term $\mu(\sigma,F)=\max\{|a_n|e^{\sigma\lambda_n}:n\ge0\}$ and central index $\nu(\sigma,F)=\max\{n\ge0:|a_n|e^{\sigma\lambda_n}=\mu(\sigma,F)\}$ are defined for all $\sigma<A$. It is proved that if $\ln\mu(\sigma,F)\le(1+o(1))\Phi(\sigma)$ as $\sigma\uparrow A$, then the inequalities $$ \varlimsup_{\sigma\uparrow A}\frac{\mu(\sigma,F')}{\mu(\sigma,F)\overline{\Phi}\,^{-1}(\sigma)}\le1,\qquad \varlimsup_{\sigma\uparrow A}\frac{\lambda_{\nu(\sigma,F')}}{\Gamma^{-1}(\sigma)}\le1, $$ hold, and these inequalities are sharp.

Growth estimates for a Dirichlet series and its derivative

Let $A\in(-\infty,+\infty]$, $\Phi$ be a continuous function on $[a,A)$ such that for every $x\in\mathbb{R}$ we have$x\sigma-\Phi(\sigma)\to-\infty$ as $\sigma\uparrow A$, $\widetilde{\Phi}(x)=\max\{x\sigma -\Phi(\sigma)\colon \sigma\in [a,A)\}$ be the Young-conjugate function of $\Phi$, $\overline{\Phi}(x)=\widetilde{\Phi}(x)/x$ for all sufficiently large $x$, $(\lambda_n)$ be a nonnegative sequence increasing to $+\infty$, $F(s)=\sum a_ne^{s\lambda_n}$ be a Dirichlet series absolutely convergent in the half-plane $\operatorname{Re}s<A$, $M(\sigma,F)=\sup\{|F(s)|\colon \operatorname{Re}s=\sigma\}$ and $G(\sigma,F)=\sum |a_n|e^{\sigma\lambda_n}$ for each $\sigma<A$. It is proved that if $\ln G(\sigma,F)\le(1+o(1))\Phi(\sigma)$, $\sigma\uparrow A$, then the inequality$$\varlimsup_{\sigma\uparrow A}\frac{M(\sigma,F')}{M(\sigma,F)\overline{\Phi}\,^{-1}(\sigma)}\le1$$holds, and this inequality is sharp. % Abstract (in English)

Generalized types of the growth of Dirichlet series

Let $A\in(-\infty,+\infty]$ and $\Phi$ be a continuously on $[\sigma_0,A)$ function such that $\Phi(\sigma)\to+\infty$ as $\sigma\to A-0$. We establish a necessary and sufficient condition on a nonnegative sequence $\lambda=(\lambda_n)$, increasing to $+\infty$, under which the equality$$\overline{\lim\limits_{\sigma\uparrow A}}\frac{\ln M(\sigma,F)}{\Phi(\sigma)}=\overline{\lim\limits_{\sigma\uparrow A}}\frac{\ln\mu(\sigma,F)}{\Phi(\sigma)},$$holds for every Dirichlet series of the form $F(s)=\sum_{n=0}^\infty a_ne^{s\lambda_n}$, $s=\sigma+it$, absolutely convergent in the half-plane ${Re}\, s<A$, where $M(\sigma,F)=\sup\{|F(s)|:{Re}\, s=\sigma\}$ and $\mu(\sigma,F)=\max\{|a_n|e^{\sigma\lambda_n}:n\ge 0\}$ are the maximum modulus and maximal term of this series respectively.

On L1-Convergence of Fourier Series under the MVBV Condition

Let f ∈ L2π be a real-valued even function with its Fourier series , and let Sn( f , x), n ≥ 1, be the n-th partial sum of the Fourier series. It is well known that if the nonnegative sequence ﹛an﹜ is decreasing and limn→∞an = 0, thenWe weaken the monotone condition in this classical result to the so-called mean value bounded variation (MVBV) condition. The generalization of the above classical result in real-valued function space is presented as a special case of the main result in this paper, which gives the L1-convergence of a function f ∈ L2π in complex space. We also give results on L1-approximation of a function f ∈ L2π under the MVBV condition.

Infinite log-concavity: developments and conjectures

International audience Given a sequence $(a_k)=a_0,a_1,a_2,\ldots$ of real numbers, define a new sequence $\mathcal{L}(a_k)=(b_k)$ where $b_k=a_k^2-a_{k-1}a_{k+1}$. So $(a_k)$ is log-concave if and only if $(b_k)$ is a nonnegative sequence. Call $(a_k)$ $\textit{infinitely log-concave}$ if $\mathcal{L}^i(a_k)$ is nonnegative for all $i \geq 1$. Boros and Moll conjectured that the rows of Pascal's triangle are infinitely log-concave. Using a computer and a stronger version of log-concavity, we prove their conjecture for the $n$th row for all $n \leq 1450$. We can also use our methods to give a simple proof of a recent result of Uminsky and Yeats about regions of infinite log-concavity. We investigate related questions about the columns of Pascal's triangle, $q$-analogues, symmetric functions, real-rooted polynomials, and Toeplitz matrices. In addition, we offer several conjectures. Étant donné une suite $(a_k)=a_0,a_1,a_2,\ldots$ de nombres réels, on définit une nouvelle suite $\mathcal{L}(a_k)=(b_k)$ où $b_k=a_k^2-a_{k-1}a_{k+1}$. Alors $(a_k)$ est log-concave si et seulement si $(b_k)$ est une suite non négative. On dit que $(a_k)$ est $\textit{infiniment log-concave}$ si $\mathcal{L}^i(a_k)$ est non négative pour tout $i \geq 1$. Boros et Moll ont conjecturé que les lignes du triangle de Pascal sont infiniment log-concave. Utilisant un ordinateur et une version plus forte de log-concavité, on vérifie leur conjecture pour la $n$ième ligne, pour tout $n \leq 1450$. On peut aussi utiliser nos méthodes pour donner une preuve simple d'un résultat récent de Uminsky et Yeats à propos des régions de log-concavité infini. Reliées à ces idées, on examine des questions à propos des colonnes du triangle de Pascal, des $q$-analogues, des fonctions symétriques, des polynômes avec racines réelles, et des matrices de Toeplitz. De plus, on offre plusieurs conjectures.

Sharp conditions for the oscillation of delay difference equations

Suppose that {pn} is a nonnegative sequence of real numbers and let k be a positive integer. We prove that limn→∞inf [1k∑i=n−kn−1pi]>kk(k+1)k+1 is a sufficient condition for the oscillation of all solutions of the delay difference equation An+1−An+pnAn−k=0, n=0,1,2,…. This result is sharp in that the lower bound kk/(k+1)k+1 in the condition cannot be improved. Some results on difference inequalities and the existence of positive solutions are also presented.