Articles of inequality

Difficult inequalities Question

Prove that if $x_i > 0$ for all $i$ then $$\begin{align*} &(x_1^{19}+x_2^{19}+\cdots+x_n^{19})(x_1^{93}+x_2^{93}+\cdots+x_n^{93})\\ &\geq (x_1^{20}+x_2^{20}+\cdots+x_n^{20})(x_1^{92}+x_2^{92}+\cdots+x_n^{92}). \end{align*}$$ and find when equality holds. I’m not sure how to start…I tried maybe using induction, but I wasn’t sure how. I also thought about using the Mean Inequality Chain and Cauchy, but I couldn’t figure out how to use it. […]

Two ways to show that $\sin x -x +\frac {x^3}{3!}-\frac {x^5}{5!}< 0$

Show that: $\large \sin x -x +\frac {x^3}{3!}-\frac {x^5}{5!}< 0$ on: $0<x<\frac {\pi}2$ I tried to solve it in two ways and got a little stuck: One way is to use Cauchy’s MVT, define $f,g$ such that $f(x)=\sin x -x +\frac {x^3}{3!}$ and $g(x)=\frac {x^5}{5!}$ so $\frac {f(x)}{g(x)}<1$ and both are continuous and differentiable on […]

Prove that $\nu(n) \le \nu(2^{n}-1)$ where $\nu(n)$ is the number of positive divisors of n

This question already has an answer here: Prove that $\tau(2^n-1) \geq \tau(n)$ for all positive integers $n$. 1 answer

Proving that the line integral $\int_{\gamma_{2}} e^{ix^2}\:\mathrm{d}x$ tends to zero

Let $f(z) = e^{iz^2}$ and $\gamma_2 = \{ z : z = Re^{i\theta}, 0 \leq \theta \leq \frac{\pi}{4} \} $. All the sources I have found online, says that the line integral $$ \left| \int_{\gamma_2} e^{iz^2}\mathrm{d}z \right| $$ tends to zero as $R \to \infty$. By using the ML-inequality one has $$ \left| \int_{\gamma_2} e^{iz^2}\mathrm{d}z […]

Prove the inequality $4S \sqrt{3}\le a^2+b^2+c^2$

Let a,b,c be the lengths of a triangle, S – the area of the triangle. Prove that $$4S \sqrt{3}\le a^2+b^2+c^2$$

Inequality for integral

Define the following integral with $n$ an integer greater than $1$: $$I_{n}=\int_{0}^{1}\frac{e^t}{(1+t)^n}dt.$$ Is it true that for all $n \geq 2$, $$ \frac{1}{n-1}\left(1-\frac{1}{2^{n-1}}\right)\leq I_{n} \leq \frac{e}{n-1}\left(1-\frac{1}{2^{n-1}}\right)?$$

How to prove that $\det(A+B) ≥ \det A +\det B$?

If $A$ and $B$ are $n \times n$ symmetric matrices with eigenvalues bigger or equal with $0$, how can I prove that $\det(A+B) \geq \det A +\det B$?

Bounding the solution of a wave equation in 3 dimensions

Let $u:{\mathbb{R}^ + } \times {\mathbb{R}^3} \to \mathbb{R}$ be a solution of the Cauchy problem $\left\{ \begin{gathered} {u_{tt}} – \Delta u = 0 \\ u\left( {0,x} \right) = {u_0}\left( x \right) \\ {u_t}\left( {0,x} \right) = {u_1}\left( x \right) \\ \end{gathered} \right.$ Assuming that ${u_0}$ and ${u_1}$ are smooth and compactly supported, show that $\left( […]

Poincaré Inequality – Product Of Measures

I’m given two euclidean spaces $ \mathbb{R}_1 , \mathbb{R}_2 $ , with probability measures on them , that satisfy the Poincaré’s inequality: $ \lambda^2 \int_{\mathbb{R}^k} |f – \int_{\mathbb{R}^k} f d\mu | ^2 d\mu \leq \int_{\mathbb{R}^k} | \nabla f | ^2 d\mu $ for some constants $C_1 , C_2 $ respectively. How can I prove that […]

A trigonometric inequality: $\cos(\theta) + \sin(\theta) > 0$

How can I find $\theta$ such that $$\cos(\theta)+\sin(\theta)>0$$