List questions from jensen-inequality

Published on 16 Nov 2019 - 10:43

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Prove that $(x + \sqrt[3]{abc})^3 \le (x + a)(x + b)(x + c) \le ( x + \frac{a + b + c}{3})^3$

Published on 21 Nov 2019 - 15:34

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How do I prove that $(\int_a^b f(x) dx)^2 \leq (b-a)\int_a^b f(x)^2 dx$?

Published on 21 Nov 2019 - 23:56

66

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Prove an inequality : $\sum_{cyc}\frac{a^3}{abu+b^2v}\geq \frac{a+b+c}{u+v}$ without Jensen's inequality

Published on 23 Mar 2026 - 0:40

141

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If $x,y>0,$ prove: $\quad x⁡Γ(y)+yΓ(x)\geq (x+y)Γ(\frac{x+y}{2})$

Published on 03 Dec 2019 - 6:57

58

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$(x+y)Γ(x+y)+(y+z)Γ(y+z)+(z+x)Γ(z+x)≥6$

Published on 05 Dec 2019 - 13:45

62

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Generalized Jensen's equality on averages

Published on 05 Dec 2019 - 17:12

101

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Jensen's inequality probability density powers and products

Published on 08 Dec 2019 - 4:27

198

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Inequality Proof $(x^2+1)(y^2+1)(z^2+1)\leq...$

Published on 10 Dec 2019 - 21:23

73

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Inequality $\sum_{cyc}a^3\frac{(16a^2-10ab+12b^2)}{(13a^2+5b^2)^2}\geq \frac{a+b+c}{18}$

Published on 18 Dec 2019 - 10:05

59

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Inequation involving summation

Published on 20 Dec 2019 - 14:53

108

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Prove that $\binom{a_1}{2} + \binom{a_2}{2} + \cdots + \binom{a_n}{2} \ge r\binom{k+1}{2} + \left(n-r\right)\binom{k}{2}$

Published on 31 Dec 2019 - 7:09

79

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minimum value of $\sum (x+1/x)^{10}$ expression

Published on 01 Jan 2020 - 14:00

238

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Inequality about the sum of square root of 1 over x

Published on 04 Jan 2020 - 6:15

99

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Showing $\sum_{k=1}^n \frac{1}{1+a_k} \ge\frac{n}{1+\sqrt[n]{\prod_{k=1}^{n} a_k}} .$

Published on 07 Jan 2020 - 19:52

List Question

Counter-example wanted for a conjecture on polynomials

Prove that $(x + \sqrt[3]{abc})^3 \le (x + a)(x + b)(x + c) \le ( x + \frac{a + b + c}{3})^3$

How do I prove that $(\int_a^b f(x) dx)^2 \leq (b-a)\int_a^b f(x)^2 dx$?

Prove an inequality : $\sum_{cyc}\frac{a^3}{abu+b^2v}\geq \frac{a+b+c}{u+v}$ without Jensen's inequality

If $x,y>0,$ prove: $\quad x⁡Γ(y)+yΓ(x)\geq (x+y)Γ(\frac{x+y}{2})$

$(x+y)Γ(x+y)+(y+z)Γ(y+z)+(z+x)Γ(z+x)≥6$

Generalized Jensen's equality on averages

Jensen's inequality probability density powers and products

Inequality Proof $(x^2+1)(y^2+1)(z^2+1)\leq...$

Inequality $\sum_{cyc}a^3\frac{(16a^2-10ab+12b^2)}{(13a^2+5b^2)^2}\geq \frac{a+b+c}{18}$

Inequation involving summation

Prove that $\binom{a_1}{2} + \binom{a_2}{2} + \cdots + \binom{a_n}{2} \ge r\binom{k+1}{2} + \left(n-r\right)\binom{k}{2}$

minimum value of $\sum (x+1/x)^{10}$ expression

Inequality about the sum of square root of 1 over x

Showing $\sum_{k=1}^n \frac{1}{1+a_k} \ge\frac{n}{1+\sqrt[n]{\prod_{k=1}^{n} a_k}} .$

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