# Tagged Questions

Questions on proving, manipulating and applying inequalities.

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### Solving $\log(x-2) + \log(9-x) \lt 1$.

The solution. Now, in comment section, a person has mentioned (and it's given in the answer behind the book) that another solution could be $2 \lt x \lt 4$. I've tried numerous times, but have not ...
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### Proving an upper bound on the terms of a sequence defined defined by a recurrence relation

Problem: Suppose $R>0$, $x_0 >0$ and $$x_{n+1}=\frac{1}{2}\left(\frac{R}{x_n}+x_n\right)$$ $n \geq 0$. Prove,$$x_n - \sqrt{R} \leq \frac{(x_0-\sqrt{R})^2}{2^nx_0}$$ for $n \geq 1$. My ...
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### How does $(\sqrt{xy}+\sqrt{yz}+\sqrt{zx})/3$ relate to $\sqrt[3]{xyz}$ for $x,y,z>0$?

This kind of 'mean' for three positive real numbers $x,y,z$ appears in some applications (elliptic integrals for example) and various inequality problems. We have, by AM-GM-HM inequalities for pairs ...
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### $\min$ value of $f(x)=a\sec x+b\csc x\;,$ Where $a,b>0$ and $\displaystyle x\in \left(0,\frac{\pi}{2}\right)$

$\min$ value of $f(x)=a\sec x+b\csc x\;,$ Where $a,b>0$ and $\displaystyle x\in \left(0,\frac{\pi}{2}\right)$ Although we can solve it using Derivative Test, But my question is can we solve it ...
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### P is any point inside a triangle . prove that s<AP+BP+CP<2s [duplicate]

P is any point inside a triangle ABC. The perimeter of the triangle AB+BC+CA=2s. Prove that s<AP+BP+CP<2s.
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### Solution of $|x^2-2x|+|x-4|>|x^2-3x+4|$

Solve the given inequality $$|x^2-2x|+|x-4|>|x^2-3x+4|$$ Obviously we can break the modulus sign for different intervals of $x$ and solve but I was wondering if any short cut exists for this ...
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### Upper bound for decreasing sequence $a_{n+1}\leq a_{n} - C a_{n+1}^2$

A monotonically decreasing sequence $(a_n)$ satisfy $a_{n+1}\leq a_{n} - C a_{n+1}^2$ where $C$ is a constant. If $a_1$ is known, could we verify that $$a_n \leq \frac{3/(2C) + a_1}{n}$$
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### Confused with a basic math problem

For what values of $x$, does the relation $\left|x + \dfrac{1}{x}\right| \le 4$ hold? I can see $x\ne0$.
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### Show that $\sum_{i=1}^{n}\left(\frac{1}{1+a_{i}}\right)^n\ge \frac{n}{2^n}$

Let $a_{1},a_{2},\cdots,a_{n}(n\ge 2)$ be postive real numbers,such that $$a_{1}a_{2}\cdots a_{n}=1$$show that $$\sum_{i=1}^{n}\left(\dfrac{1}{1+a_{i}}\right)^n\ge \dfrac{n}{2^n}$$ In fact,the ...
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### On a second-order differential inequality involving the Dirichlet eta function

After that I've tried understand the problem 6416 [1983, 60] A Second-Order Differential Inequality proposed by Sandford S. Miller in the American Mathematical Monthly (myself proposal is ...
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### Solution of $\sqrt{x+i}+\frac{(i+x)^2}{1-i} \geq 9$

Solve the following inequality: $$\sqrt{x+i}+\frac{(i+x)^2}{1-i} \geq 9$$ Does this inequality makes sense? I could see that $L.H.S.$ should be real number too otherwise there won't be any ...
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### Question on Waring's Problem

I've been confused on the nuances between the inequality found with the study of Waring's problem on the Wikipedia page "Floor and ceiling functions" versus the standard "Waring's Problem" page. On "...
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### Proving some inequalities related to Information Theory

I've been working on some inequalities related to the information theory section of my decision theory course, and I could use some help on some of the derivations for one of the inequalities. As a ...
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$x,y,z >0$ and $x+y+z=3$, prove $$\tag{1}\frac{xy}{5y^3+4}+\frac{yz}{5z^3+4}+\frac{zx}{5x^3+4} \leqslant \frac13$$ My first attempt is to use Jensen's inequality. Hence I consider the function $... 0answers 101 views ### Prove$\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y} \geqslant \frac32+ \frac{27}{16}\frac{(y-z)^2}{(x+y+z)^2}x,y,z >0$, prove $$\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y} \geqslant \frac32+ \frac{27}{16}\frac{(y-z)^2}{(x+y+z)^2}$$ This inequality is easier than the other one. Previously, I learned ... 1answer 96 views ### Prove$\frac{x+y+z}{3}+\frac{3}{\frac1x+\frac1y+\frac1z}\geq5\sqrt[3]{\frac{xyz}{16}}$Let$x,y,z>0$. Prove that $$\frac{x+y+z}{3}+\frac{3}{\frac1x+\frac1y+\frac1z}\geq5\sqrt[3]{\frac{xyz}{16}}$$ On the left-hand side we have arithmetic and harmonic means, while on the right ... 0answers 39 views ### Solving an exponential inequality problem How do I prove the following inequality : $$\Bigg(\frac{2}{\alpha^2} \, \big( e^{\alpha x} - e^{\alpha y} \big) \, + \, e^{\alpha y} (y^2 - x^2) \; \Bigg) > 0$$ given,$x, y > 0$? Can ... 0answers 14 views ### Prove that$K(u) = ||u||Mu+f$is a contraction Let$M$be a$2 \times 2$real matrix such that $$\parallel{}Mx\parallel{} \leq \frac{1}{4}\parallel{}x\parallel{}$$ for all$x \in \mathbb{R}^2$where$\parallel{}\parallel{}$is the euclidean norm ... 1answer 211 views ### Prove$\sqrt[2]{x+y}+\sqrt[3]{y+z}+\sqrt[4]{z+x} <4x,y,z \geqslant 0$and$x^2+y^2+z^2+xyz=4$, prove $$\sqrt[2]{x+y}+\sqrt[3]{y+z}+\sqrt[4]{z+x} <4$$ A natural though is that from the condition$x^2+y^2+z^2+xyz=4$, I tried a trig substitutions ... 3answers 527 views ### Prove that$\frac{x^x}{x+y}+\frac{y^y}{y+z}+\frac{z^z}{z+x} \geqslant \frac32x,y,z >0$, prove $$\frac{x^x}{x+y}+\frac{y^y}{y+z}+\frac{z^z}{z+x} \geqslant \frac32$$ I have a solution for this beautiful and elegant inequality. I am posting this inequality to share and ... 0answers 25 views ### Triangle inequality Trace norm when becomes the triangle inequality for the trace norm an equality? I search for it in books and web, but couldn´t find it. Thanks for help! 4answers 68 views ### Solving inequalities involving square roots What is the correct way to solve inequalities like$\sqrt {x+2}>x $? I deduced that$x$must be greater than$-2$and also on squaring I get$-1 \leq x \leq 2$but even on combining these two I'... 3answers 84 views ###$a$and$b$be positive real numbers such that$a + b = 1$. Prove that$a^a \cdot b^b+a^b\cdot b^a≤1$Let$a$and$b$be positive real numbers such that$a + b = 1$. Prove that $$a^a\cdot b^b+a^b \cdot b^a≤1$$ 1answer 41 views ### Showing that$\text{exp}(\left \lvert z \right \rvert/(\left\lvert z \right\rvert - 1))\leq \left \lvert 1 + z \right\rvert$Suppose$z \in \mathbb{C}$with$\left\lvert{z}\right\rvert < 1$. I want to prove that that $$\exp\left(\frac{|z|}{|z| - 1}\right)\leq |1 + z|.$$ I tried writing$z$in cartesian form as$x + iy$... 1answer 117 views ###$(x+2y+z)\cdot \left( \frac{x}{y} +\frac{2y}{z}+\frac{z}{x}\right) > 12$for$x^2+y^2+z^2=3x,y,z > 0$and$x^2+y^2+z^2=3$, prove $$(x+2y+z)\cdot \left( \frac{x}{y} +\frac{2y}{z}+\frac{z}{x}\right) > 12$$ The coefficient$2$destroys the symmetry of this inequality and makes the ... 1answer 116 views ### Prove$(x+y+z) \cdot \left( \frac1x +\frac1y +\frac1z\right) \geqslant 9 + \frac{4(x-y)^2}{xy+yz+zx}x,y,z >0$, prove $$(x+y+z) \cdot \left( \frac1x +\frac1y +\frac1z\right) \geqslant 9 + \frac{4(x-y)^2}{xy+yz+zx}$$ The term$\frac{4(x-y)^2}{xy+yz+zx}$made this inequality tougher. It remains me ... 4answers 62 views ### How to prove that$\frac {e^{b^2-1}}{b^2}\$ ≥ 1
How to prove that $$\frac {e^{b^2-1}}{b^2} \ge 1?$$ Use logarithm or limit or what? Or do we have to use it as a conclusion to prove it backwards? And how to prove it forwards, that is, without ...