# Tagged Questions

Questions on proving, manipulating and applying inequalities.

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### Lower bound on the sum of singular values for a sum of Hermitian matrices

Suppose $\mathbf{A}$ is a Hermitian $n\times n$ matrix with eigenvalues $\lambda_i(\mathbf{A})$, $i=1,\ldots,n$. Suppose $\mathbf{B}$ is an $n \times n$ complex-valued matrix and $b\neq 0$ is a ...
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### Series and integrals for inequalities and approximations to $\pi$

Fundamentals Two beautiful expressions that relate $\pi$ to its convergents are Dalzell integral $$\frac{22}{7}-\pi=\int_0^1\frac{x^4(1-x)^4}{1+x^2}dx$$ (see Why do we need an integral to prove ...
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### Proving weak coercivity by young's and interpolation inequalities

Let be $$(P)\left\{\begin{array}{ll} &-\Delta u + V(x)u=f & \text{ in }\ \Omega\\ &u=0 & \text{ on } \ \Gamma \end{array}\right.$$ with $V \in L^r(\Omega)$, for some ...
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### Eigenvalues of $MA$ versus eigenvalues of $A$ for orthogonal projection $M$

Suppose that $M$ is symmetric idempotent $n\times n$ and has rank $n-k$. Suppose that $A$ is $n\times n$ and positive definite. Let $0<\nu_1\leq\nu_2\leq\ldots\nu_{n-k}$ be the nonzero ...
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### L2 Norm of Inverse of Non-square Matrix Multiplication

Consider a matrix $A\in\mathbb R^{n\times m}$ with $n<m$. Given that $\|A\|_2 = \gamma_0$ and $AA^T$ is invertible, can we find any equality/upper bound for $\|(AA^T)^{-1}\|_2$ in terms of ...
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### Proving an inequality involving integrals?

I am trying to prove that $$[\sum_{i=1}^{n}(\ln t_i)^2 t_i^\alpha+A^{\prime \prime}(\alpha)][\sum_{i=1}^{n}t_i^\alpha+A(\alpha)]\ge[\sum_{i=1}^{n}(\ln t_i) t_i^\alpha+A^{\prime}(\alpha)]^2$$ where ...
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### In order to show or refute: Given a real function $f$ and $a, b \in R$ then $a\leq b \Rightarrow f(a)\leq f(b)$, what should I regard?

Is it enough to show that $f$ is increasing or decreasing in any interval $I$ that contains both numbers $a$ and $b$?
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### A series to prove $\frac{22}{7}-\pi>0$

After T. Piezas answered Is there a series to show $22\pi^4>2143\,$? a natural question is Is there a series that proves $\frac{22}{7}-\pi>0$? One such series may be found combining ...
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### To clear for variable 'a' in a sum of dependent products

I can't seem to find a way to clear this equation for variable $a$: $E[k] = \displaystyle\sum_{k=1}^nk\frac{a}{n+a-k}\displaystyle\prod_{i=0}^{k-1}1-\frac{a}{n+a-i}$ Do you think it's possible? Any ...
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### Inequality involving exponential function (base $2$ and $3$) [on hold]

Show that the following inequality holds for every real number x: $$3^x+0.5>2^x$$
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### L2 Matrix Norm Upper Bound in terms of Bounds of its Column

I need to find an upper bound for a matrix norm in terms of bounds of its columns. I have a vector $\varepsilon_i(x) \in R^{n\times1}$ such that $||\varepsilon_i(x)||_2\le\gamma_0$. I also have a ...
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### Investigating Nicolas' criterion for the Riemann Hypothesis. [on hold]

Throughout this note, $N_k$ denotes the $k$-th primorial number (the product of the first $k$ primes), $\varphi(n)$ the Euler totient function, and $\gamma$ is the Euler-Mascheroni constant. By the ...
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### Proof for $x\le -1 \implies x^3-x\le 0$?

Here is my proof: Let $x\in \mathbb{R}$, assume $x\le -1$ Then $x^2\ge 1$ Then $x^3\le -1$ Since $x\le -1$ $x^3\le x$ Then $x^3-x\le 0$ Therefore $x\le -1 \implies x^3-x\le 0$ Therefore ...
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### Basic question $|x^2| < 9$

I have a rather basic question. Let's assume that $|x^2| < 9$, where $x\in \mathbb{R}$. Then everyone knows that $x \in$ (-3,3). However, I have trouble arriving at the answer based on basic ...
### $a+\frac{1}{a}\ge 2$ for $a\in\mathbb{R}_{+}$
This inequality is more than obvious: $$a+\frac{1}{a}\ge 2$$ But my question is: is this only a special case of some "bigger" lemma (like e.g. $\frac{a+b}{2}\ge\sqrt{ab}$ is a special ase of the ...