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Prove that $|z^2+1|\le 2$ implies $|z^3+3z+2|\le 6$

May 16

Without calculator prove that $9^{\sqrt{2}} < \sqrt{2}^9$

Jan 4, 2023

If a two variable smooth function has two global minima, will it necessarily have a third critical point?

Feb 20, 2021

Find the real root of the almost symmetric polynomial $x^7+7x^5+14x^3+7x-1$

Oct 23, 2022

Show $\frac{\sin x_1\sin x_2\cdots\sin x_n}{\sin(x_1+x_2)\sin(x_2+x_3)\cdots\sin(x_n+x_1)}\le\frac{\sin^n(\pi/n)}{\sin^n(2\pi/n)}$, for $\sum x_i=\pi$

Jun 19, 2019

Questions

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About the inequality $x^{x^{x^{x^{x^x}}}} \ge \frac12 x^2 + \frac12$

Jun 7, 2021

The limit and asymptotic analysis of $a_n^2 - n$ from $a_{n+1} = \frac{a_n}{n} + \frac{n}{a_n}$

Aug 24, 2020

Prove that $a^{4/a} + b^{4/b} + c^{4/c} \ge 3$

Jul 28, 2022

Prove/Disprove $|1 + z_1| + |1 + z_2| + |1 + r z_1 z_2| \ge 1 + \min(r, 1/r)$

Jul 21, 2022

Prove that $16(a\sin a + \cos a - 1)^2 \le 2a^4 + a^3 \sin 2a, \ \forall a\ge 0$

Jan 29, 2021

Reputation

+10

Show $f(x_1,x_2) = \frac{1}{x_1x_2}$ is convex for $(x_1,x_2) \in \mathbb{R}^2_{++}$

+15

Conjecture: For $a,b,c,A,s$ the sides, area, and semi-perimeter of a triangle, and $x\ge2$: $a+b-c\ge s(2/3)^{(x+1)/(x-1)}(\frac{3^{3/2}A}{s^2})^x$

+10

Prove that $|z^2+1|\le 2$ implies $|z^3+3z+2|\le 6$

+10

Is $U^{-1}(U(x)+U(y))$ a convex function in general?

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