Articles of inequality

Prove an inequality by Induction: $(1-x)^n + (1+x)^n < 2^n$

Could you give me some hints, please, to the following problem. Given $x \in \mathbb{R}$ such that $|x| < 1$. Prove by induction the following inequality for all $n \geq 2$: $$(1-x)^n + (1+x)^n < 2^n$$ $1$ Basis: $$n=2$$ $$(1-x)^2 + (1+x)^2 < 2^2$$ $$(1-2x+x^2) + (1+2x+x^2) < 2^2$$ $$2+2x^2 < 2^2$$ $$2(1+x^2) < 2^2$$ […]

Points in unit square

Let $n$ points be given in the unit square. How to prove or disprove: the points can be labeled $x_1,\ldots,x_n$ to satisfy the inequality $$\|x_1-x_2\|^2 +\|x_2-x_3\|^2+\cdots+\|x_n-x_1\| ^2 \le 4,$$ where $\|\cdot\| $ is the Euclidian distance?

Proof of Ptolemy's inequality?

Can anyone prove the Ptolemy inequality, which states that for any convex quadrulateral $ABCD$, the following holds:$$\overline{AB}\cdot \overline{CD}+\overline{BC}\cdot \overline{DA} \ge \overline{AC}\cdot \overline{BD}$$ I know this is a generalization of Ptolemy’s theorem, whose proof I know. But I have no idea on this one, can anyone help?

Alternate Proof for $e^x \ge x+1$

This is just a standard problem from my high school’s calculus text, but my proof seems sort of off. This is it: Let $f(x) = e^x$. The tangent line of $f(x)$ at $x=0$ is $g(x)=x+1$. Since $f”(x_0) \gt 0$ for all $x_0 \in \mathbb R$,the tangent line $g(x) \le f(x)$ for all $x$. Q.E.D. I […]

Prove the inequality $\frac 1a + \frac 1b +\frac 1c \ge \frac{a^3+b^3+c^3}{3} +\frac 74$

Inequality Let $a$, $b$ and $c$ be positive numbers such that $a+b+c=3$. Prove the following inequality $$\frac 1a + \frac 1b +\frac 1c \ge \frac{a^3+b^3+c^3}{3} +\frac 74.$$ I stumbled upon this question some days ago and been trying AM-GM to find the solution but so far have been unsuccessful.

Square Root Inequality

How can I prove the following inequality: Given $ a,b>0 $ and $a^2>b $, we have $a>\sqrt b$ Thank you.

Young's inequality without using convexity

I was doing some problems from Rudin’s Principles of Mathematical Analysis and came across a problem in which he asks you to prove Hölder’s inequality via Young’s inequality: If $u$ and $v$ are nonnegative real numbers, and $p$ and $q$ are positive real numbers such that $\displaystyle \frac{1}{p}+\frac{1}{q}=1$, then $\displaystyle uv \leq \frac{1}{p}u^p+\frac{1}{q}v^q$. I’m familiar […]

Prove inequality $\sqrt{\frac{1}n}-\sqrt{\frac{2}n}+\sqrt{\frac{3}n}-\cdots+\sqrt{\frac{4n-3}n}-\sqrt{\frac{4n-2}n}+\sqrt{\frac{4n-1}n}>1$

For any $n\ge2, n \in \mathbb N$ prove that $$\sqrt{\frac{1}n}-\sqrt{\frac{2}n}+\sqrt{\frac{3}n}-\cdots+\sqrt{\frac{4n-3}n}-\sqrt{\frac{4n-2}n}+\sqrt{\frac{4n-1}n}>1$$ My work so far: 1) $$\sqrt{n+1}-\sqrt{n}>\frac1{2\sqrt{n+0.5}}$$ 2) $$\sqrt{n+1}-\sqrt{n}<\frac1{2\sqrt{n+0.375}}$$

Show that $2 < e^{1/(n+1)} + e^{-1/n}$

I’m trying to show $2 < e^{1/(n+1)} + e^{-1/n}$. I can show that $ 2 < e^{1/n} + e^{-1/(n+1)}$ since $$2 \leq 2\cosh\left(\frac{1}{n}\right) = e^{1/n} + e^{-1/n} < e^{1/n} + e^{-1/(n+1)}$$ but I’m still having trouble with the other inequality. I though using $\cosh$ again might help but I can’t get anywhere.

To show that $e^x > 1+x$ for any $x\ne 0$

This question already has an answer here: Simplest or nicest proof that $1+x \le e^x$ 21 answers