# Questions tagged [inequalities]

for questions involving inequalities.

**3**

votes

**3**answers

140 views

### A challenging inequality that involves the digamma function and polygamma functions

Let $f(x)=x \psi(x+1)$, where $\psi$ is the digamma function. Define
$$g(x)=(f(ax)+f((1-a)x)-f(x))-(f(ax+by)+f((1-a)x+(1-b)y)-f(x+y)),$$
where $0\le a,b\le 1$ and $x,y\ge 0$.
How to show that $g(x)$ ...

**1**

vote

**0**answers

123 views

### Lower bound for the $L^2$ norm of a polynomial / hypergeometric function

Let $\sigma >0$ be fixed. For even $k \in \mathbb{N} \cup \{0\}$, we consider the polynomial
\begin{equation}
\varphi_k(x) = \sum_{j=0}^{k} (-1)^j {k \choose j} b_j \, x^{2j} \quad x \in (-1,1),
\...

**1**

vote

**1**answer

55 views

### Hanson-Wright inequality (quadratic form concentration inequality) for bounded random vectors

Is there a concentration inequality for quadratic forms of bounded random vectors $X \in [-1, 1]^n$ with zero mean and given covariance matrix $\Sigma \in \mathbb{R}^{n \times n}$ but otherwise ...

**8**

votes

**1**answer

561 views

### A curious inequality concerning binomial coefficients

Has anyone seen an inequality of this form before? It seems to be true (based on extensive testing), but I am not able to prove it.
Let $a_1,a_2,\ldots,a_k$ be non-negative integers such that $\sum_i ...

**0**

votes

**0**answers

57 views

### Does there exist $\alpha>0, \beta\in (0,1)$ such that $\dfrac{\sum_{k=1}^n a_k}{n}\le \alpha (a_1\cdots a_n)^{1/n} + \beta \max_i(a_i)$ holds?

Let $a_1\ge a_2\ge \cdots\ge a_n\ge 0$ be given non-negative numbers. My question is the following:
Is there any $\beta \in(0,1),\ \alpha>0$, such that $$\dfrac{a_1+\cdots+a_n}{n}\le \alpha (a_1\...

**1**

vote

**1**answer

225 views

### Prove that $x^{y^x}>y^{x^y}$ for $x>y>0.$ [closed]

Let $x>y>0$. Prove that $$x^{y^x}>y^{x^y}.$$
My attempts:
Let $1>x>y>0$.
In this case it's enough to prove that $$y^x<x^y$$ or
$$x\ln\frac{1}{y}>y\ln\frac{1}{x},$$ which is ...

**2**

votes

**1**answer

207 views

### An isoperimetric inequality for curve in the plane?

Let $f(x,y)=0$ be a (smooth) simple closed curve $C$ on the plane and $R$ the region bounded by $C$ (appropriately oriented). Assume the origin lies in the interior of $R$.
QUESTION. Let $r=\sqrt{x^...

**1**

vote

**2**answers

134 views

### Infimum upper bound

Let $h:[0,1]\to[0,1]$ be a $\mathcal{C^1}$ function such that $h'(x)<0$ for all $x\in(0,1)$. I am trying to show that (not sure if it is true):
$$
\inf_{f\in \mathcal{H}}\left(\int_0^1\int_0^1 h(|...

**0**

votes

**0**answers

78 views

### Definition for Norm of a Tensor [on hold]

I am reading a book on differential geometry and at one point the following statement is made:
$|\text{Ric} + \text{Hess}(f)|^2 \geq \frac{1}{n}|R+\Delta f|^2.$
The justification is that 'we have ...

**5**

votes

**1**answer

141 views

### Binomial Distributions and Inequality

Suppose $c>2$ is a real number, and $x$ solves equation $f(\frac{1}{2}+x,m,2m)=(2m+1)cx$ for some $m>0$ integer number, where $f$ is probability mass function for binomial distribution, with ...

**6**

votes

**1**answer

140 views

### An inequality for rearrangement-style sums

The following is a holdover from my math contest days that I never got around
to solve.
We will use the notation $\left[ k\right] $ for the set $\left\{
1,2,\ldots,k\right\} $ whenever $k$ is a ...

**2**

votes

**2**answers

156 views

### An inequality on elementary symmetric polynomial of eigenvalues

For $A \in \mathbb{R}^{n\times n}$, I am wondering if the following inequality holds,
$$\text{tr}(A)^2 - \text{tr}(A^2) \leq ||A||_*^2 - ||A||_F^2$$
This is equivalent to the following inequality on ...

**0**

votes

**1**answer

64 views

### Inequality involving product-of-minus vs minus-of-product for positive integers

I'm encountering this inequality for dimensionality reduction problem. The simplified form looks as follows:
Consider positive integers $a_1$, $a_2$, $b_1$ and $b_2$ where $a_1>b_1$ and $a_2>...

**12**

votes

**3**answers

2k views

### Does anyone recognize this inequality?

In some paper the authors make use of the following inequality without further explanation: Let $x\in\mathbb{R}^n$ with $x_1\le\cdots\le x_n$ and $\alpha\in[0,1]^n$ with $\sum_{i=1}^n \alpha_i=N\in\{1,...

**1**

vote

**1**answer

58 views

### Good UPPER bounds for $\log(\sum_{i=1}^n p_ie^{z_i})-\sum_{i=1}^np_iz_i$ where $(p_i)_i$ is a probability vector

Let $x=(z_1,\ldots,z_n)$ be real vector and $(p_1,\ldots,p_n)$ be a probability vector.
Question
$\log(\sum_{i=1}^n p_ie^{z_i})-\sum_{i=1}^np_iz_i \le ???$
Observation
This paper allows us to ...