Question

Use the fact that $\ln(1+x) \leq x$ to show that $(a_1a_2\cdot\cdot\cdot a_n)^{1/n} \leq \frac{a_1+a_2+\cdot\cdot\cdot+a_n}{n}$

score 101 · Answer 1 · 2018-06-30 15:05:12

101

Views

Use the fact that $\ln(1+x) \leq x$ to show that $(a_1a_2\cdot\cdot\cdot a_n)^{1/n} \leq \frac{a_1+a_2+\cdot\cdot\cdot+a_n}{n}$

Published on 30 Jun 2018 - 15:05

score 151 · Answer 2 · 2026-03-25 04:38:22

151

Views

What tools one should use for inequalities?

Published on 25 Mar 2026 - 4:38

score 88 · Answer 3 · 2018-07-04 00:21:14

88

Views

Showing that $x+y+z \le 2\left(\frac {x^2}{y+z}+ \frac {y^2}{x+z}+ \frac {z^2}{x+y}\right)$ for positive $x, y, z$?

Published on 04 Jul 2018 - 0:21

score 124 · Answer 4 · 2018-07-06 01:35:52

124

Views

For $x\geq 0$, what is the smallest value of $\frac{4x^2+8x+13}{6(x+1)}$?

Published on 06 Jul 2018 - 1:35

score 155 · Answer 5 · 2018-07-08 09:58:31

155

Views

Using Method of Undetermined coefficients to find maximum value

Published on 08 Jul 2018 - 9:58

score 243 · Answer 6 · 2018-07-13 23:39:29

243

Views

Intermediate inequalities; is there a way to know if you're getting a "bad deal"?

Published on 13 Jul 2018 - 23:39

score 102 · Answer 7 · 2018-07-16 17:16:55

102

Views

An inequality with substitution

Published on 16 Jul 2018 - 17:16

score 178 · Answer 8 · 2018-07-16 19:01:43

178

Views

Prove: $\log_2(x)+\log_x(y)+\log_y(8)\geq \sqrt[3]{81}$

Published on 16 Jul 2018 - 19:01

score 170 · Answer 9 · 2018-07-17 18:51:33

170

Views

Show $\frac{1}{b+c+d} + \frac{1}{a+c+d} + \frac{1}{a+b+d} + \frac{1}{a+b+c} \ge \frac{16}{3(a+b+c+d)}$.

Published on 17 Jul 2018 - 18:51

score 75 · Answer 10 · 2018-07-18 17:07:11

75

Views

How do I prove the following inequality?

Published on 18 Jul 2018 - 17:07

score 260 · Answer 11 · 2026-03-25 04:43:22

260

Views

$ \frac {3 x y z}{(x+y) (y+z) (z+x)} + \sum\limits_{cycl}^{} \left(\frac {x+y}{x+y+ 2 z}\right)^2 \ge \frac {9}{8}.$

Published on 25 Mar 2026 - 4:43

score 206 · Answer 12 · 2018-07-18 21:57:34

206

Views

Showing that if $p_1 + \cdots p_n = 1$ then $\displaystyle \sum_{k=1}^n \left(p_k + \dfrac {1}{p_k} \right)^2 \ge n^3+2n+\dfrac 1n$?

Published on 18 Jul 2018 - 21:57

score 242 · Answer 13 · 2018-07-20 13:15:46

242

Views

Prove: $\log_2(x)+\log_3(x)+\log_5(x)>9\log_{30}(x)$

Published on 20 Jul 2018 - 13:15

score 413 · Answer 14 · 2026-03-25 12:48:45

413

Views

Prove that $a^4+b^4+1\ge a+b$.

Published on 25 Mar 2026 - 12:48

score 102 · Answer 15 · 2018-07-24 07:51:54

102

Views

Algebraic Inequality involving AM-GM-HM

Published on 24 Jul 2018 - 7:51

List Question

Use the fact that $\ln(1+x) \leq x$ to show that $(a_1a_2\cdot\cdot\cdot a_n)^{1/n} \leq \frac{a_1+a_2+\cdot\cdot\cdot+a_n}{n}$

What tools one should use for inequalities?

Showing that $x+y+z \le 2\left(\frac {x^2}{y+z}+ \frac {y^2}{x+z}+ \frac {z^2}{x+y}\right)$ for positive $x, y, z$?

For $x\geq 0$, what is the smallest value of $\frac{4x^2+8x+13}{6(x+1)}$?

Using Method of Undetermined coefficients to find maximum value

Intermediate inequalities; is there a way to know if you're getting a "bad deal"?

An inequality with substitution

Prove: $\log_2(x)+\log_x(y)+\log_y(8)\geq \sqrt[3]{81}$

Show $\frac{1}{b+c+d} + \frac{1}{a+c+d} + \frac{1}{a+b+d} + \frac{1}{a+b+c} \ge \frac{16}{3(a+b+c+d)}$.

How do I prove the following inequality?

$ \frac {3 x y z}{(x+y) (y+z) (z+x)} + \sum\limits_{cycl}^{} \left(\frac {x+y}{x+y+ 2 z}\right)^2 \ge \frac {9}{8}.$

Showing that if $p_1 + \cdots p_n = 1$ then $\displaystyle \sum_{k=1}^n \left(p_k + \dfrac {1}{p_k} \right)^2 \ge n^3+2n+\dfrac 1n$?

Prove: $\log_2(x)+\log_3(x)+\log_5(x)>9\log_{30}(x)$

Prove that $a^4+b^4+1\ge a+b$.

Algebraic Inequality involving AM-GM-HM

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