Question

How do I show this $xy\leq \frac{1}{p}x^p+\frac{1}{q}y^q.$ without using concavity of log function?

score 590 · Answer 1 · 2026-04-16 06:42:37

590

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How do I show this $xy\leq \frac{1}{p}x^p+\frac{1}{q}y^q.$ without using concavity of log function?

Published on 16 Apr 2026 - 6:42

score 57 · Answer 2 · 2026-04-17 06:13:35

57

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Prove $\int_Rfg\,dm\leq\|f\|_p^{1-p/r}\|g\|_p^{1-q/r}(\int_Rf^pg^q\,dm)^{1/r}$, where $1\leq p\leq\infty$ and $\frac1{r}=\frac1{p}+\frac1{q}-1$

Published on 17 Apr 2026 - 6:13

score 169 · Answer 3 · 2026-04-12 18:47:02

169

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Inequality with $(x+y)(y+z)(z+w)(w+x)=1$

Published on 12 Apr 2026 - 18:47

score 64 · Answer 4 · 2026-04-17 05:08:20

64

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A Cauchy-Schwarz-type inequality for $\int\prod_n|f_n|$

Published on 17 Apr 2026 - 5:08

score 95 · Answer 5 · 2026-04-10 22:49:41

95

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Interchanging limit and integral.

Published on 10 Apr 2026 - 22:49

score 224 · Answer 6 · 2026-05-10 14:36:20

224

Views

Prove that $\frac{a}{c\sqrt{a^2+1}}+\frac{b}{a\sqrt{b^2+1}}+\frac{c}{b\sqrt{c^2+1}}\ge \frac{3}{2}$

Published on 10 May 2026 - 14:36

score 191 · Answer 7 · 2026-03-25 04:57:02

191

Views

Alternative form for Liapunov inequality

Published on 25 Mar 2026 - 4:57

score 590 · Answer 8 · 2026-04-16 09:09:28

590

Views

Use of Holder inequality in gradient estimate for harmonic function.

Published on 16 Apr 2026 - 9:09

score 67 · Answer 9 · 2026-04-10 23:59:52

67

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Doubt in variant of Holder inequality

Published on 10 Apr 2026 - 23:59

score 136 · Answer 10 · 2026-04-17 02:35:54

136

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Cauchy Schwartz Inequality Question: $(a^2+b^2)^3=c^2+d^2 \implies \frac{a^3}{c}+\frac{b^3}{d}\geq 1$

Published on 17 Apr 2026 - 2:35

score 164 · Answer 11 · 2026-04-13 02:17:24

164

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Trig Integral Inequality

Published on 13 Apr 2026 - 2:17

score 160 · Answer 12 · 2026-05-10 14:31:25

160

Views

Show that $a^{2014}+b^{2014}\geq a^{2013}+b^{2013} $.

Published on 10 May 2026 - 14:31

score 458 · Answer 13 · 2026-04-12 06:12:31

458

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Showing that an integral operator on $L^p$ spaces has a certain norm

Published on 12 Apr 2026 - 6:12

score 256 · Answer 14 · 2026-04-16 18:07:16

256

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$\frac{1}{3}\sum_{cyc}\frac{1}{\sqrt{1+a}}\ge\frac{1}{\sqrt{1+\sqrt[3]{abc}}}$ if $\;a, b, c\;$ are positive reals s.t. $abc\ge2^9$

Published on 16 Apr 2026 - 18:07

score 261 · Answer 15 · 2026-04-15 17:55:56

261

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A lower bound for Holder's inequality and an upper bound for the reverse Holder inequality!

Published on 15 Apr 2026 - 17:55

List Question

How do I show this $xy\leq \frac{1}{p}x^p+\frac{1}{q}y^q.$ without using concavity of log function?

Prove $\int_Rfg\,dm\leq\|f\|_p^{1-p/r}\|g\|_p^{1-q/r}(\int_Rf^pg^q\,dm)^{1/r}$, where $1\leq p\leq\infty$ and $\frac1{r}=\frac1{p}+\frac1{q}-1$

Inequality with $(x+y)(y+z)(z+w)(w+x)=1$

A Cauchy-Schwarz-type inequality for $\int\prod_n|f_n|$

Interchanging limit and integral.

Prove that $\frac{a}{c\sqrt{a^2+1}}+\frac{b}{a\sqrt{b^2+1}}+\frac{c}{b\sqrt{c^2+1}}\ge \frac{3}{2}$

Alternative form for Liapunov inequality

Use of Holder inequality in gradient estimate for harmonic function.

Doubt in variant of Holder inequality

Cauchy Schwartz Inequality Question: $(a^2+b^2)^3=c^2+d^2 \implies \frac{a^3}{c}+\frac{b^3}{d}\geq 1$

Trig Integral Inequality

Show that $a^{2014}+b^{2014}\geq a^{2013}+b^{2013} $.

Showing that an integral operator on $L^p$ spaces has a certain norm

$\frac{1}{3}\sum_{cyc}\frac{1}{\sqrt{1+a}}\ge\frac{1}{\sqrt{1+\sqrt[3]{abc}}}$ if $\;a, b, c\;$ are positive reals s.t. $abc\ge2^9$

A lower bound for Holder's inequality and an upper bound for the reverse Holder inequality!

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