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15 votes
2 answers
788 views

Combinatorial inequality involving alternating signs

I would like to prove the following inequality. It arises from my study of random matrices. I have verified the inequality for $q\in \{0.01,0.02, \ldots, 0.99\}$ and $1\le n\le 100$. Let $n$ be any ...
shortfatboy's user avatar
9 votes
2 answers
354 views

Asymptotics of a quadratic recursion

Consider the sequence defined by \begin{align} c_0 &{}= 1 \\ c_n &{}= 2\,n\,c_{n-1}-\frac{1}{2}\sum_{m=1}^{n-1}c_m\,c_{n-m}. \end{align} How can you prove that it has the following asymptotics ...
Matteo Beccaria's user avatar
2 votes
1 answer
276 views

Combinatorial optimization problem on sums of differences between real numbers

We are given an increasing sequence $S$ of positive real numbers $x_1, x_2, \ldots, x_n$, such that $$x_{i+2}-x_{i+1} \ge c\,(x_{i+1}-x_i)$$ for all $i=1, 2, \ldots n-2$, where $c\ge 1$ is constant. ...
Penelope Benenati's user avatar
1 vote
0 answers
105 views

Does this inequality follow from doubly log-concave?

On a sequence $(a_k)_{k\geq0}$ of positive integers, define the operator $\mathcal{L}a_k=a_k^2-a_{k-1}a_{k+1}$. Then, $(a_k)_k$ is called log-concave if $\mathcal{L}a_k\geq0$ for all $k\geq0$. One may ...
T. Amdeberhan's user avatar
0 votes
1 answer
60 views

Bounding the ratio of the $\ell_1$-norms of two real-valued $n$-vectors as a linear combination of their $n$ element-wise ratios

Let $a_1, a_2, \ldots a_n$ and $b_1, b_2, \ldots b_n$ be two sequences of $n\gg 1$ real numbers such that, for all $1\le i\le n$, we have $0<a_i \le b_i\le 1$. Let the ratio $R$ defined as follows: ...
Penelope Benenati's user avatar