query: If $\left|z^3 + {1 \over z^3}\right| \le 2$ then $\left|z + {1 \over z}\right| \le 2$
positive_passages.docid: 17519
positive_passages.text: Let $z \in\Bbb{C}^{\times}$ such that $|z^3+\frac{1}{z^3}|\leq 2$. Prove that $|z+\frac{1}{z}|\leq 2$.
negative_passages.docid: 45742
negative_passages.text: Prove that ${\left(\frac{x+1}{2}\right)}^{x+1} \leq x^x$
negative_passages.docid: 113250
negative_passages.text: If $x+{1\over x} = r $ then what is $x^3+{1\over x^3}$?
negative_passages.docid: 6169
negative_passages.text: If $ A^3=A$ prove that $Ker\left(A-I\right)+Im\left(A-I\right)=V$
negative_passages.docid: 64277
negative_passages.text: How to prove ${{a}^{a}}{{b}^{b}}\ge {{\left(\frac{a+b}{2}\right)}^{a+b}}$ ?thanks.
negative_passages.docid: 101371
negative_passages.text: show that $f^{(3)}(c) \ge 3$ for $c\in(-1,1)$
negative_passages.docid: 13794
negative_passages.text: $|z_{1} + z_{2}| = |z_{1} - z_{2}| \implies z_{1}/z_{2}$ is Imaginary
negative_passages.docid: 60665
negative_passages.text: Meaning of $\mathbb{Z}\left[\sqrt{3}\right]$?
negative_passages.docid: 94516
negative_passages.text: If $3$ divides $a^2 + b^2$, then $3$ divides $a$ and $3$ divides $b$
negative_passages.docid: 84313
negative_passages.text: $|f(z)|\le 1-|z|\forall z\in D$, we need to show $f\equiv 0$
negative_passages.docid: 34017
negative_passages.text: Show that $n^{\pi\left(2n\right)-\pi\left(n\right)}2$
negative_passages.docid: 89429
negative_passages.text: Prove $\sup \left| f'\left( x\right) \right| ^{2}\leqslant 4\sup \left| f\left( x\right) \right| \sup \left| f''\left( x\right) \right| $
negative_passages.docid: 158647
negative_passages.text: If $|f(z)|\le |f(z^2)|$ then prove that $f$ is constant
negative_passages.docid: 4315
negative_passages.text: show that if $n\geq1$, $(1+{1\over n})^n<(1+{1\over n+1})^{n+1}$
negative_passages.docid: 82961
negative_passages.text: Show that $\left| \frac{1}{x}-\frac{1}{z} \right| \le \left|\frac{1}{x} - \frac{1}{y} \right| + \left|\frac{1}{y} - \frac{1}{z} \right|$.
negative_passages.docid: 21921
negative_passages.text: Showing that $|f(z)| \leq \prod \limits_{k=1}^n \left|\frac{z-z_k}{1-\overline{z_k}z} \right|$
negative_passages.docid: 44562
negative_passages.text: If $\left|z+\frac{2}{z}\right |=2$, then prove that max value of $|z|$ is $\sqrt 3 + 1$ (Corrected version)