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Questions tagged [inequality]

Questions on proving, manipulating and applying inequalities.

4
votes
0answers
155 views
+50

Proving $a^{ab}+b^{ab}\leq a^{a^2}+b^{b^2}$ and an identity .

I was working on this Prove that $a^{ab}+b^{bc}+c^{cd}+d^{da} \geq \pi$ when I have discovered the following identity : $$\Bigg|\Big(\frac{1}{2}\Big)^{\frac{x}{8}}\pm\Big(\frac{1}{4}\Big)^{\frac{x}{8}}...
5
votes
1answer
207 views
+50

Maximum of $\int_a^b \frac{f(x)}{x}\,\mathrm dx$

Let $b>a>0$ and $M>0$ be fixed. Let $F$ be the set of all functions $f:[a,b]\to[-M,M]$ such that $$\int_a^bf(x)\,\mathrm dx=0.$$Find$$\max_{f\in F}\int_a^b\frac{f(x)}x\,\mathrm dx.$$ I tried ...
23
votes
3answers
664 views
+150

Prove that $a^{ab}+b^{bc}+c^{cd}+d^{da} \geq \pi$

If $a,b,c,d >0$, and $a+b+c+d=4$, prove that $$a^{ab}+b^{bc}+c^{cd}+d^{da} \geq \pi.$$ I don't think Jensen's inequality will help here, but I think first determining where equality holds will ...