Articles of inequality

Is $|f(a) – f(b)| \leqslant |g(a) – g(b)| + |h(a) – h(b)|$? when $f = \max\{{g, h}\}$

Let $f = \max\{{g, h}\}$ where all 3 of these functions map $\mathbb{R}$ into itself. Is it true that $|f(a) – f(b)| \leqslant |g(a) – g(b)| + |h(a) – h(b)|$? I’m thinking it can be proven by cleverly adding and subtracting inside of the absolute value and then using the triangle inequality, but i’m completely […]

Clarkson type inequality

Is it true that for $p\in (1,2)$ the following inequalities holds: $$ 2^{p-1} (|x|^p+|y|^p)\leq |x+y|^p+|x-y|^p \leq 2 (|x|^p+|y|^p)$$ for $x, y \in \mathbb{R}$ ? Thanks.

Distance between the product of marginal distributions and the joint distribution

Given a joint distribution $P(A,B,C)$, we can compute various marginal distributions. Now suppose: \begin{align} P1(A,B,C) &= P(A) P(B) P(C) \\ P2(A,B,C) &= P(A,B) P(C) \\ P3(A,B,C) &= P(A,B,C) \end{align} Is it true that $d(P1,P3) \geq d(P2,P3)$ where d is the total variation distance? In other words, is it provable that $P(A,B) P(C)$ is a better […]

Solving an inequality involving factorials and an exponent

I wish to solve the following inequality. $\frac{(n!)^2}{(2n)!}$ $\leqslant$ $\frac{1}{500}$ $\forall$ n $\in$ $N_+$ Any help would be appreciated! Edit: I am not allowed a calculator for this question.

If $n$ is a positive integer, Prove that $\frac1{2^2}+\frac1{3^2}+\dotsb+\frac1{n^2}\lt\frac{2329}{3600}.$

If $n$ is a positive integer, Prove that $$\frac1{2^2}+\frac1{3^2}+\dotsb+\frac1{n^2}\lt\frac{2329}{3600}.$$ please don’t refer to the famous $1+\frac1{2^2}+\frac1{3^2}+\dotsb=\frac{\pi^2}6$. I am looking a method that doesn’t use $\text{“}\pi\text{”}.$ Unfortunately, I know and tried only $\text{“}\pi\text{”}$ method.

Convergence of a recursively defined sequence

Let a sequence $(a_n)_{n=0}^\infty$ be defined recursively $a_{n+1} = (1-a_n)^{\frac1p}$, where $p>1$, $0<a_0<(1-a_0)^{\frac1p}$. Let $a$ be the unique real root of $a=(1-a)^{\frac1p}$, $0<a<1$. It is clear $0<a_0<(1-a_0)^{\frac1p}\Leftrightarrow 0<a_0<a$. Prove 1) $a_{2k-2}<a_{2k}<a<a_{2k+1}<a_{2k-1}$ and $a_{2k+1}-a<a-a_{2k}$. 2) $\lim\limits_{n\to\infty}a_n=a$. Define $f(x):=(1-x)^{\frac1p}$. Consider $f^2$. When $p=2$, $a_{n+2}=f^2(a_n)=\big(1-(1-a_n)^{\frac12}\big)^{\frac12}$. $a_{n+2}>a_n\Leftrightarrow (1-a_n)(1+a_n)^2>1\Leftrightarrow a_n<(1-a_n)^{\frac12}$, and the conclusion is proved. But I am having difficulty […]

How to prove that $\frac x{\sqrt y}+\frac y{\sqrt x}\ge\sqrt x+\sqrt y$

I am trying to prove that $$\frac x{\sqrt y}+\frac y{\sqrt x}\ge\sqrt x+\sqrt y$$ I tried some manipulations, like multiplications in $\sqrt x$ or $\sqrt y$ or using $x=\sqrt x\sqrt x$, but I’m still stuck with that. What am I missing?

Generalized Poincaré Inequality on H1 proof.

let’s see if someone can help me with this proof. Let $\Omega\subset\mathbb{R}^n$ be a bounded domain. And let $L^2\left(\Omega\right)$ be the space of equivalence classes of square integrable functions in $\Omega$ given by the equivalence relation $u\sim v \iff u(x)=v(x)\, \text{a.e.}$ being a.e. almost everywhere, in other words, two functions belong to the same equivalence […]

Show that $1/\sqrt{1} + 1/\sqrt{2} + … + 1/\sqrt{n} \leq 2\sqrt{n}-1$

This question already has an answer here: How to prove the inequality $2\sqrt{n + 1} − 2 \le 1 +\frac 1 {\sqrt 2}+\frac 1 {\sqrt 3}+ \dots +\frac 1 {\sqrt n} \le 2\sqrt n − 1$? 3 answers

How to show: $\sin \leq t\sin x+(1-t)\sin y,$ where $0\leq t\leq 1,$ and $\pi\leq x, y\leq 2\pi$.

How can I show that $$\sin [tx+(1-t)y]\leq t\sin x+(1-t)\sin y$$ where $0\leq t\leq 1$ and $\pi\leq x, y\leq 2\pi$? I know I have to use convex function property, but I am not able to prove this inequality.