here is what I need to proof, have no idea were to start. I know there is some connection with the Stirling theorem. $$ \sum_{i=0}^{d}\binom{m}{i} \leq \left ( \frac{em}{d} \right …

I need to bound a sum of a portion of binomial coefficients in terms of "the next one", and understand what is the best which can be said in this sense. Given a real number $t \ge …

Question. Given a positive-definite $n \times n$ matrix $A = (a_{ij})$ with eigenvalues $$ \lambda_1 \leq \cdots \leq \lambda_n , $$ is there a sharp upper bound for the product $ …

In its asymmetric version, the Mahler conjecture states that if $K \subset \mathbb{R^n}$ is a convex body containing the origin as an interior point and $$ K^* := \{y \in \mathbb{ …

Hi, For a given $\theta < 1$, and $N$ a positive integer, I am trying to find an $x > 0$ (preferably the smallest such $x$) such that the following inequality holds: $$\sum_{k …

While analyzing convergence speed of stochastic-gradient methods for convex optimization problems, Recht et al (2011) posed a tantalizing conjecture. It seems quite tricky, so afte …

The following problem has a beautiful geometric interpretation in terms of the proportion of points on the Euclidean sphere in $\mathbb{R}^d$ that lie at least a certain distance a …

Does anyone know of any tight upper/lower bound for incomplete Gamma functions? i.e either of the following functions: $$ \Gamma(s,x) = \int_x^{\infty} t^{s-1}\,e^{-t}\,{\rm d}t …

Hi, I am wondering what one can say regarding the convex combination of i.i.d Gamma random variables? Specifically, consider $x_{i}$ be $Gamma(\theta,1)$, then would we have the …

For $u\in W^{k,p}(U)$, where $U\subseteq\mathbb{R}^n$ is open and bounded with $C^1$-boundary, we have the celebrated Sobolev inequalities: If $k < n/p$ then $u\in L^q(U)$ for $ …

Consider $N$ i.i.d random variables, $X_{1}, X_{2}, \ldots, X_{N}$ , that are chi-squared of degree $K \geq 2$. Also consider the following 3 vectors: \begin{eqnarray*} \bar{a} &a …

Short version Gibbs's inequality is a simple inequality for real numbers, usually understood information-theoretically. In the jargon, it states that for two probability measures …

I wish to show that $|j_n(x)| < \frac{1}{\sqrt{x}}$ for $n=0,1,2,\ldots$ and $x>0$, where $j_n$ is the spherical Bessel function of the first kind. Experimenting with Matlab …

Consider the Chernoff-Hoeffding bound, stated as follows: Let $X_1, \dots, X_K$ be i.i.d. real-valued random variables with expectation value $\mu$ and satisfying $|X_i| \le b$. Le …

For all $s>0$ define for $\epsilon\in(0,1)$ the function: \begin{equation} g(\epsilon)=\sum_{k=0}^{\infty}(1+k)^s(\sqrt{1-\epsilon})^k. \end{equation} Prove that $\exists C>0$ and …