# Questions tagged [inequality]

Questions on proving, manipulating and applying inequalities. Do not use this tag just because an inequality appears somewhere in your question.

24,997 questions
Filter by
Sorted by
Tagged with
27 views

### Prove $\sum_{k=1}^{n-1}\frac{(n-k)^2}{2k} \geq \frac{n^2\log(n)}{8}$

As the title says, prove $$\sum_{k=1}^{n-1}\frac{(n-k)^2}{2k} \geq \frac{n^2\log(n)}{8},$$ for $n>1$. This inequality is from Erdős, "Problems and results on the theory of interpolation". ...
68 views

86 views

### Prove or disprove: $7 \lt \sqrt{3} + \sqrt{27}$ [duplicate]

In an admission test to enroll in a Earth's Science Bachelor Degree course there is this question: Sort in increasing order $7$, $\sqrt{47}$ and $\sqrt{3} + \sqrt{27}$. Now, I know that $7=\sqrt{49}$...
12 views

### Upper bound by diagnoal matrix

I got stuck in proving the following inequality. Let $Y∈R^{n×p}$ with nonzero singular values $(n>p)$, $Λ=diag(λ_1,λ_2,⋯,λ_n)$ with $λ_1>λ_2>⋯>λ_n>0$. $e_n=[0,0,⋯,0,1]^⊤∈R^n$, \begin{...
30 views

28 views

### Condition for 1 to be between the roots of a function.

Q. Find the range of value of x for which 1 lies between the roots of the equation. $3y^2-(3sinx)y -2cos^2x=0$ By IVT, we know that $f(1)<0$, $3y^2-(3sin1)y-2cos^2(1)=0$ should have roots which are ...
60 views

### I have an equation I need to prove but I don't really have an idea how to: [closed]

$$\left||y|\,x^2-|b|\,a^2\right| \leq 2(|x-a|+|y-b|), \quad\forall x,y,a,b \in(-1,1)$$ It is part of a much longer proof and I'm not entirely sure that it is necessary (or reasonable).
43 views

### how to prove this inequality equation?

I am trying to prove the following inequality for $R>1$ R is a constant: $$\Bigl(1-\frac{1}{n+1}\Bigr)^R\ge1-\frac{R}{n}$$ I am not sure if it's correct or not, but I tried to prove it with ...
29 views

### Does the following inequality hold - the inner product divided by the product of norms?

Let $\cdot$ denotes the dot product and $||\boldsymbol{x}||$ denotes the $L^2-$norm of the vector $\boldsymbol{x}$. Suppose $\boldsymbol{a,b,c}$ are vectors in $\mathbb{R}^3$. Does the following ...
59 views

### Two conjectures about the prime counting function : $\frac{\pi{(x)}}{\pi{(\pi{(x)}})}<\ln(x)$

Let $x\geq 100$ then we have as conjecture : $$\frac{\pi{(x)}}{\pi{(\pi{(x)}})}<\ln(x)$$ I have tested at $x=100$ to $x=5000000000$ without any counter-example. The first fact : It seems that the ...
40 views

26 views

### why the largest value of $k$ for any number of the form $𝑚^𝑘<B$ is $\lfloor\log B \rfloor$

Given an integer $n>2$, let $B=\lceil\log^5n\rceil$. I am trying to understand why the largest value of $k$ for any number of the form $m^k \le B$, $m\ge2$ is $\lfloor\log B \rfloor$. Logarithms ...
19 views

### what are 3 solutions to the problem: x ≤ 50 [closed]

the boy cannot answer more than 50 questions
26 views

17 views

### Inequalities: How to show $W_t \ge M_t$ for $t \in [0, T]$?

$\mu$ and $\lambda$ are known and $\mu > \lambda > 0$. For $t \in [0, T]$, we have \begin{align} \frac{dW_t}{dt} = \mu W_t - \mu H_t, H_t > W_t , W_T = 0 \\ \frac{dM_t}{dt} = \lambda M_t - \...
20 views

### In a C*-algebra , if $a\leq b$, then $a^{\alpha}\leq b^{\alpha}$, for $0<\alpha\leq1$ [closed]
I need help to find a reference to prove the the following theorem: Theorem 6.9. Let $A$ be a $C^*$ algebra. If $a,b\in A_+$ and $a\leq b$ then $a^\alpha\leq b^\alpha$ for $0\leq \alpha\leq 1$. On ...