Articles of inequality

Prove that if $a,b,$ and $c$ are positive real numbers, then $\frac{a^3}{a^3+2b^3}+\frac{b^3}{b^3+2c^3}+\frac{c^3}{c^3+2a^3} \geq 1.$

Prove that if $a,b,$ and $c$ are positive real numbers, then $$\dfrac{a^3}{a^3+2b^3}+\dfrac{b^3}{b^3+2c^3}+\dfrac{c^3}{c^3+2a^3} \geq 1.$$ This question seems hard since we aren’t given any other information about $a,b,c$. I think expanding it might got somewhere, but I don’t see Cauchy-Schwarz or AM-GM leading anywhere.

How can I prove that $x-{x^2\over2}<\ln(1+x)$

How can I prove that $$\displaystyle x-\frac {x^2} 2 < \ln(1+x)$$ for any $x>0$ I think it’s somehow related to Taylor expansion of natural logarithm, when: $$\displaystyle \ln(1+x)=\color{red}{x-\frac {x^2}2}+\frac {x^3}3 -\cdots$$ Can you please show me how? Thanks.

Prove $\sum\limits_{cyc} \frac{\sqrt{xy}}{\sqrt{xy+z}}\le\frac{3}{2}$ if $x+y+z=1$

if $x,y,z$ are positive real numbers and $x+y+z=1$ Prove:$$\sum_{cyc} \frac{\sqrt{xy}}{\sqrt{xy+z}}\le\frac{3}{2}$$ where $\sum_{cyc}$ denotes sums over cyclic permutations of the symbols $x,y,z$. Additional info:I’m looking for solutions and hint that using Cauchy-Schwarz and AM-GM because I have background in them. Things I have done so far: Using AM-GM $$xy+z \ge 2$$ $$\sqrt{xy+z} \ge \sqrt2$$ So manipulating […]

Why is $x^{1/n}$ continuous?

Why is $x^{1/n}$ continuous for positive $x,n$ where $n$ is an integer? I can’t see how it follows from the definition of limit. And I don’t see any suitable inequalities so is this an application of Bernoulli’s or Jensen’s inequality?

$\sup\{g(y):y\in Y\}\leq \inf\{f(x):x\in X\}$

Let $X$ and $Y$ be two nonempty sets and let $h:X\times Y\rightarrow \mathbb{R}$ have a bounded range in $\mathbb{R}$.Let $f:X\rightarrow \mathbb{R}$ and $g:Y\rightarrow \mathbb{R}$ defined by $$f(x)=\sup\{h(x,y):y\in Y\}$$ and $$g(y)=\inf\{h(x,y):x\in X\}$$Then can we prove that $$\sup\{g(y):y\in Y\} \leq \inf\{f(x):x\in X\}?$$

Inequality constant in Papa Rudin

The following inequality appears in Rudin’s Real and Complex Analysis, 3ed, in the proof of ($f$) in Theorem 9.2 (Fourier Transforms): if $x\in\mathbb{R}$ and $\phi(x,u):= (e^{-ixu} – 1)/u$ then $|\phi(x,u)|\le |x|$ for all real $u\ne 0$. It seems to me that this is incorrect and that the inequality should read “$\le 2|x|$”. Can anyone please […]

Does $1 + \frac{1}{x} + \frac{1}{x^2}$ have a global minimum, where $x \in \mathbb{R}$?

Does the function $$f(x) = 1 + \frac{1}{x} + \frac{1}{x^2},$$ where $x \in {\mathbb{R} \setminus \{0\}}$, have a global minimum? I tried asking WolframAlpha, but it appears to give an inconsistent result.

Proving that $\phi_a(z) = (z-a)/(1-\overline{a}z)$ maps $B(0,1)$ onto itself.

I want to prove that if $\phi_a: B(0,1) \to \Bbb C$ is given by $\phi_a(z) = (z-a)/(1-\overline{a}z)$ with $|a| < 1$, then $|\phi_a(z)| < 1$. Resist the itch on your finger urging you to close the question: I already took a look at this question and this one. I’m supposed to prove things in the […]

Find $\Big\{ (a,b)\ \Big|\ \big|a\big|+\big|b\big|\ge 2/\sqrt{3}\ \text{ and }\forall x \in\mathbb{R}\ \big|a\sin x + b\sin 2x\big|\le 1\Big\}$

Find all (real) numbers $a $ and $b$ such that $|a| + |b| \ge 2/\sqrt{3} $ and for any $x$ the inequality $|a\sin x + b \sin 2x | \le 1$ holds. In other words, find the set $Q$ defined as $$Q = \Big\{\ (a,b)\ \Big| \quad 1. \left|a\right| + \left|b\right| \ge \frac{2}{\sqrt{3}}, \ \text{ […]

Does convergence in probability preserve the weak inequality?

Suppose I have two sequences of random variables $\{x_n\}$ and $\{y_n\}$ such that $x_n\leq y_n$ and $\text{plim}\;x_n=L_x$ and $\text{plim}\;y_n=L_y$, can I say $L_x\leq L_y$ (almost surely)? Does it matter if I further impose that $L_x$ and $L_y$ are nonrandom? I tried to replicate the argument for the nonstochastic case (included below for completeness) but I […]