Intereting Posts

gcd of $(2^{2^m} + 1 , 2^{2^n}+1) = 1$ for distinct pair of positive integers $n,m$
Countable compact spaces as ordinals
If $f$ is a meromorphic modular form of weight $k$, then $\frac{1}{f}$ is a modular form of weight $-k$
Formalize a proof without words of the identity $(1 + 2 + \cdots + n)^2 = 1^3 + 2^3 + \cdots + n^3$
Is this graph connected
How to find a linearly independent vector?
Is linear algebra more “fully understood” than other maths disciplines?
Solving functional equation $f(x)f(y) = f(x+y)$
How to calculate $f(x)$ in $f(f(x)) = e^x$?
Derivative of conjugate transpose of matrix
Logical squabbles
Why is this allowed? (“Fourier's Trick”; finding the coefficients in a Fourier Series)
Do these integrals have a closed form? $I_1 = \int_{-\infty }^{\infty } \frac{\sin (x)}{x \cosh (x)} \, dx$
Why algebraic topology is also called combinatorial topology?
Is $\lim_{n\to \infty} \frac{np_n}{\sum_{i=1}^n p_i} = 2$ true?

I could use some help with proving this inequality:

$$\left|\,x_1\,\right|+\left|\,x_2\,\right|+…+\left|\,x_p\,\right|\leq\sqrt{p}\sqrt{x^2_1+x^2_2+…+x^2_p}$$ for all natural numbers p.

Aside from demonstrating the truth of the statement itself, apparently $\sqrt{p}$ is the smallest possible value by which the right hand side square root expression must be multiplied by in order for the statement to be true. I’ve tried various ways of doing this, and I’ve tried to steer clear of induction because I’m not sure that’s what the exercise was designed for (from Bartle’s Elements of Real Analysis), but the best I’ve been able to come up with is proving that the statement is true when the right hand side square root expression is multiplied by p, which seems pretty obvious anyway. I feel like I’m staring directly at the answer and still can’t see it. Any help would be appreciated.

- To show that $P(|X-Y| \leq 2) \leq 3P(|X-Y| \leq 1)$
- Olympiad inequality $\frac{a}{2a + b} + \frac{b}{2b + c} + \frac{c}{2c + a} \leq 1$.
- Given $abc(a+b+c)=3$ prove $(a+b)(b+c)(c+a)>8$
- Prove $\frac{ab}{1+c^2}+\frac{bc}{1+a^2}+\frac{ca}{1+b^2}\le\frac{3}{4}$ if $a^2+b^2+c^2=1$
- Show that if $X \succeq Y$, then $\det{(X)}\ge\det{(Y)}$
- Find minimum of $a+b$ under the condition $\frac{m^2}{a^2}+\frac{n^2}{b^2}=1$ where $m,n$ are fixed arguments

- Simple Integral $\int_0^\infty (1-x\cot^{-1} x)dx=\frac{\pi}{4}$.
- Continuous Functions and Cauchy Sequences
- If every real-valued continuous function is bounded on $X$ (metric space), then $X$ is compact.
- Real-analytic $f(z)=f(\sqrt z) + f(-\sqrt z)$?
- Show that ${n \choose k}\leq n^k$
- Prime powers, patterns similar to $\lbrace 0,1,0,2,0,1,0,3\ldots \rbrace$ and formulas for $\sigma_k(n)$
- Compute $\lim_{x\to 0}\dfrac{\sqrt{\cos x}-\sqrt{\cos x}}{x^2}$
- Second derivative positive $\implies$ convex
- Prove lower bound $\sum\limits_{k=1}^{n}\frac{1}{\sqrt{n^2+k^2}}\ge\left(1-\frac{1}{n}\right)\ln{(1+\sqrt{2})}+\frac{\sqrt{2}}{2n}$
- Relationship between diameter and radius of a point set

I imagine two possible ways to solve this: with the help of the QM-AM inequality, and with Cauchy-Schwarz inequality.

Using the QM-AM inequality, we see that:

$$\frac{|x_1|+|x_2| +… + |x_p|}{p} \le \sqrt{\frac{|x_1|^2+|x_2|^2 +… + |x_p|^2}{p}}$$

Multiplying both sides by $p$:

$$\begin{align}|x_1|+|x_2| +… + |x_p| &\le p\sqrt{\frac{|x_1|^2+|x_2|^2 +… + |x_p|^2}{p}}\\

&=\sqrt{p}\sqrt{|x_1|^2 + |x_2|^2 +… + |x_p|^2}\\

&=\sqrt{p}\sqrt{x_1^2 + x_2^2 +… + x_p^2}\end{align}$$

proving the desired statement.

Using the alternative Cauchy-Schwarz inequality (which really is, in some sense, a generalization of the QM-AM-GM-HM inequalities), we get :

$$(|x_1|\cdot1 + |x_2|\cdot1 +…+|x_p|\cdot1)^2 \le (|x_1|^2 + |x_2|^2 +… + |x_p|^2)(\underbrace{1 + 1 + … + 1}_\text{$p$})$$

or:

$$(|x_1|+|x_2| +… + |x_p|)^2 \le (|x_1|^2 + |x_2|^2 +… + |x_p|^2)(p)$$

Taking square roots of both sides, we get :

$$|x_1|+|x_2| +… + |x_p| \le \sqrt{p}\sqrt{|x_1|^2 + |x_2|^2 +… + |x_p|^2}$$

But since we know that for all real $x$, $|x|^2 = x^2$, we can reduce this to:

$$|x_1|+|x_2| +… + |x_p| \le \sqrt{p}\sqrt{x_1^2 + x_2^2 +… + x_p^2}$$

This is an application of Jensen’s Inequality:

$$

\left(\frac1p\sum_{k=1}^p|x_k|\right)^2\le\frac1p\sum_{k=1}^p|x_k|^2

$$

since $f(x)=x^2$ is convex.

- Riemann integrals of abstract functions into Banach spaces
- Integral of differential form and integral of measure
- seeing the differential dx/y on an elliptic curve as an element of the sheaf of differentials
- Show that $\lim\limits_{x\to0}\frac{e^x-1}{x}=1$
- If $\forall x \in R, x^2-x \in Z(G)$, than $R$ is commutative
- Computing: $L =\lim_{n\rightarrow\infty}\left(\frac{\frac{n}{1}+\frac{n-1}{2}+\cdots+\frac{1}{n}}{\ln(n!)} \right)^{{\frac{\ln(n!)}{n}}} $
- Proof of $k {n\choose k} = n {n-1 \choose k-1}$ using direct proof
- intrinsic proof that the grassmannian is a manifold
- The Biharmonic Eigenvalue Problem with Dirichlet Boundary Conditions on a Rectangle
- Where can I learn about the lattice of partitions?
- What does a standalone $dx$ mean?
- Factorial Moment of the Geometric Distribution
- how to prove $m^{\phi(n)}+n^{\phi(m)}\equiv 1 \pmod{mn}$ where m and n are relatively prime?
- How to calculate relative error when true value is zero?
- What is an example of real application of cubic equations?