Question

Increasing function $f: \mathbb{N} \to \mathbb{R}^{\ge 0}$ such that $f(3^n) = 2^n$ and $\forall n \exists c : f(n+c)+f(n-c) \ge 2f(n).$

score 48 · Answer 1 · 2020-07-10 03:57:49

48

Views

Increasing function $f: \mathbb{N} \to \mathbb{R}^{\ge 0}$ such that $f(3^n) = 2^n$ and $\forall n \exists c : f(n+c)+f(n-c) \ge 2f(n).$

Published on 10 Jul 2020 - 3:57

score 115 · Answer 2 · 2020-07-19 15:04:11

115

Views

How do I find all functions $F$ with $F(x_1) − F(x_2) \le (x_1 − x_2)^2$ for all $x_1, x_2$?

Published on 19 Jul 2020 - 15:04

score 69 · Answer 3 · 2020-07-21 21:48:54

69

Views

Let $0\leq a \leq b \leq 1$. Then we have for all natural numbers $m\geq 2$ the inequality $b^{\frac m2}-a^{\frac m2} \leq\frac m2(b-a)$

Published on 21 Jul 2020 - 21:48

score 50 · Answer 4 · 2026-03-21 22:45:56

50

Views

How does $af\left(\frac{n}{b}\right) \leq cf(n)$ imply that $a^{i}f\left(\frac{n}{b^{i}}\right) \leq c^{i}f(n)$?

Published on 21 Mar 2026 - 22:45

score 69 · Answer 5 · 2026-03-22 02:03:13

69

Views

Existence of a functional inequality

Published on 22 Mar 2026 - 2:03

score 261 · Answer 6 · 2026-03-22 15:11:51

261

Views

How to show this equality for operator norm?

Published on 22 Mar 2026 - 15:11

score 418 · Answer 7 · 2016-07-21 21:13:22

418

Views

Inequality: product of integrals

Published on 21 Jul 2016 - 21:13

score 359 · Answer 8 · 2026-03-21 17:44:59

359

Views

An inequality in positive real continuous function

Published on 21 Mar 2026 - 17:44

score 130 · Answer 9 · 2026-03-22 08:06:02

130

Views

Find $f$ such that $(f(x))^2 \ge f(x + y)(f(x) + y), \forall x, y$

Published on 22 Mar 2026 - 8:06

score 368 · Answer 10 · 2026-03-22 07:19:57

368

Views

Show that: $nf(\frac{x_1+\cdots+x_n}{n})+n\left(\frac{f(M)+f(m)}{2}-f(\frac{M+m}{2})\right) \ge f(x_1)+\cdots+f(x_n)$

Published on 22 Mar 2026 - 7:19

score 229 · Answer 11 · 2026-03-22 01:05:39

$\frac{1}{2}(\frac{b_1}{a_1}-\frac{b_n}{a_n})^2(\sum_{1}^{n}{a_i^2 }) ^2 \ge (\sum_{1}^{n}{a_i^2 }) (\sum_{1}^{n}{b_i^2 })-(\sum_{1}^{n}{a_ib_i })^2$

score 621 · Answer 12 · 2026-03-21 16:17:13

621

Views

Prove inequality $\ln \left( \frac{e-e^x}{1-x} \right) \leq \sqrt{\frac{1+x+x^2}{3}}$ for $0<x \le 1$

Published on 21 Mar 2026 - 16:17

score 2.1k · Answer 13 · 2026-03-22 13:13:22

2.1k

Views

Proving the Kantorovich inequality by another inequality

Published on 22 Mar 2026 - 13:13

score 247 · Answer 14 · 2026-03-21 17:42:02

247

Views

Which matrix satisfies the following condition?

Published on 21 Mar 2026 - 17:42

score 176 · Answer 15 · 2026-03-21 22:00:28

176

Views

Counterexample or proof for the following matrix inequality?

Published on 21 Mar 2026 - 22:00

List Question

Increasing function $f: \mathbb{N} \to \mathbb{R}^{\ge 0}$ such that $f(3^n) = 2^n$ and $\forall n \exists c : f(n+c)+f(n-c) \ge 2f(n).$

How do I find all functions $F$ with $F(x_1) − F(x_2) \le (x_1 − x_2)^2$ for all $x_1, x_2$?

Let $0\leq a \leq b \leq 1$. Then we have for all natural numbers $m\geq 2$ the inequality $b^{\frac m2}-a^{\frac m2} \leq\frac m2(b-a)$

How does $af\left(\frac{n}{b}\right) \leq cf(n)$ imply that $a^{i}f\left(\frac{n}{b^{i}}\right) \leq c^{i}f(n)$?

Existence of a functional inequality

How to show this equality for operator norm?

Inequality: product of integrals

An inequality in positive real continuous function

Find $f$ such that $(f(x))^2 \ge f(x + y)(f(x) + y), \forall x, y$

Show that: $nf(\frac{x_1+\cdots+x_n}{n})+n\left(\frac{f(M)+f(m)}{2}-f(\frac{M+m}{2})\right) \ge f(x_1)+\cdots+f(x_n)$

$\frac{1}{2}(\frac{b_1}{a_1}-\frac{b_n}{a_n})^2(\sum_{1}^{n}{a_i^2 }) ^2 \ge (\sum_{1}^{n}{a_i^2 }) (\sum_{1}^{n}{b_i^2 })-(\sum_{1}^{n}{a_ib_i })^2$

Prove inequality $\ln \left( \frac{e-e^x}{1-x} \right) \leq \sqrt{\frac{1+x+x^2}{3}}$ for $0<x \le 1$

Proving the Kantorovich inequality by another inequality

Which matrix satisfies the following condition?

Counterexample or proof for the following matrix inequality?

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