Questions on proving, manipulating and applying inequalities.

learn more… | top users | synonyms (1)

0
votes
1answer
45 views

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 ...
1
vote
1answer
22 views

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 ...
1
vote
1answer
40 views

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 ...
1
vote
1answer
33 views

$\min$ 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 ...
2
votes
1answer
87 views

If $a+b+c=0$ what is the value of $\frac{a^2}{2a^2 +bc }+\frac{b^2}{2b^2 +ca }+\frac{c^2}{2c^2 +ab }$

Let $s=\frac{a^2}{2a^2 +bc }+\frac{b^2}{2b^2 +ca }+\frac{c^2}{2c^2 +ab }$. If we use inequality $\frac{x^2}{a}+\frac{y^2}{b} \ge \frac{(x+y)^2}{(a+b)}$ we get $s \ge 0$ as $a+b+c=0$. Again $s \le \...
4
votes
2answers
112 views

Any integral or series to prove $\frac{1}{\sqrt{3}}>\gamma$?

I recently noticed that these two numbers are remarkably close: $$\frac{1}{\sqrt{3}}-\gamma=0.000135\dots$$ Are there any integrals or series which can prove that $\frac{1}{\sqrt{3}}>\gamma$? ...
1
vote
1answer
75 views

Does this inequality hold?

I am currently writting a paper and ended up with an expression that looks like the following: $$\frac{\sum_{i=1}C_{i}}{\sum_{i=1} {N_i}^\gamma {C_{i}}^{1-\gamma}}$$ for $i=1...K$, $C_i>0$, $N_i ...
0
votes
0answers
20 views

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.
1
vote
2answers
41 views

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 ...
1
vote
1answer
28 views

If $a_{n+1} \leq \left(1-\frac{2}{n+1}\right)a_n + b\left(\frac{2}{n+1}\right)^2$ then $a_n \leq \frac{4b}{n+1}$

A sequence $(a_n)$ satisfy $a_{n+1} \leq (1-\gamma_n)a_n + \frac{\beta R^2}{2}\gamma_n^2$ where $\gamma_n = \frac{2}{n+1}$, $\beta$ and $R$ are constant. How to verify that $$a_n \leq \frac{2\beta R^...
1
vote
0answers
45 views

Prove an inequation about x,y,z [closed]

Suppose x,y,z are positive real number,prove $$\frac{xy}{z}+\frac{yz}{x}+\frac{zx}{y}>2\sqrt[3]{x^3+y^3+z^3}$$
0
votes
1answer
45 views

show that $x_{1}+x_{2}<1$

Given $m\in\mathbb{R}$, assume that the equation $x\ln{x}=m$ has two real roots $x_1,x_2$. Show that: $$x_{1}+x_{2}<1.$$ My attempt. Since $$x_{1}\ln{x_{1}}=x_{2}\ln{x_{2}}=m$$ if we let $$f(x)=...
8
votes
2answers
115 views

An interesting inequality $\sum_{k=1}^n \frac{1}{n+k}<\frac{\sqrt{2}}{2}, \ n\ge1$

Here is one of the beautiful inequalities from Elementary inequalities by Mitrovic $$\sum_{k=1}^n \frac{1}{n+k}<\frac{\sqrt{2}}{2},$$ which is easy to prove by calculus using that $\lim_{n\to\...
2
votes
3answers
51 views

Solve the equation $|2x-1| -|x+5| = 3$

Problem : Solve the equation $|2x-1| - |x+5| = 3$ In my attempt to solve the problem, I only manage to get one of the solutions. Attempted Solution $$\begin{equation} \begin{split} |x|-|y| & \...
1
vote
1answer
19 views

Can $\sigma(2^r)$ be abundant for $r > 1$?

Let $\sigma(x)$ be the sum of the divisors of the positive integer $x$. If $\sigma(y) < 2y$, $y$ is called deficient; if $\sigma(z) > 2z$, $z$ is called abundant. Questions (1) Can $\...
0
votes
2answers
58 views

prove this $\sqrt{2}-1\le 4\sin{\frac{A}{2}}\sin{\frac{B}{2}}\sin{\frac{C}{2}}$

In any acute triangle $\Delta ABC,min{(A,B,C)}\ge\dfrac{\pi}{4}$,Prove that $$\sqrt{2}-1\le 4\sin{\dfrac{A}{2}}\sin{\dfrac{B}{2}}\sin{\dfrac{C}{2}}$$ since $$4\sin{\dfrac{A}{2}}\sin{\dfrac{B}{2}}\sin{...
1
vote
1answer
34 views

Prove the inequality involving exponential function in form of $\exp( \frac{1}{x} )$

For $\nu > 0$, $0 < x \leq \nu $, and a positive integer $S$, (we think) following an inequality always holds $1- \left( \frac{1}{x+1} \right)^S \geq \exp \left( -\frac{1}{Sx} \right) $ Does ...
7
votes
4answers
233 views

Minimize $-\sum\limits_{i=1}^n \ln(\alpha_i +x_i)$

While solving PhD entrance exams I have faced the following problem: Minimize the function $f(x)=- \sum_{i=1}^n \ln(\alpha_i +x_i)$ for fixed $\alpha_i >0$ under the conditions: $\sum_{i=1}^n ...
0
votes
1answer
35 views

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 $$a_n \leq \frac{3/(2C) + a_1}{n}$$
0
votes
3answers
130 views

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 $x\ne0$.
5
votes
1answer
286 views

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 ...
2
votes
1answer
73 views

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 ...
-1
votes
0answers
26 views

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 ...
-2
votes
1answer
42 views

How to prove $(\sum_{k=1}^na_kb_k)^2\le \sum_{k=1}^na_k^2b_k$ for $\sum_{k=1}^nb_k=1$ using Cauchy Swarz inequality? [closed]

I have no idea like these aren't even vectors so how am I suppose to solve them with something that's meant for vectors. Let $a_k,b_k \geq 0$, with $\sum_{k=1}^nb_k=1$. Prove that $(\sum_{k=1}^...
0
votes
1answer
36 views

Proving an inequality involving a strictly convex function

Given, $f$ is a strictly convex function. Based on what assumptions on '$x$' and '$y$', can I say that the following inequality stands true : $$f(x) \; + f(y) \; > \; f(x + y) \; \; ?$$
1
vote
2answers
80 views

Extreme values of $\frac{(a+b+c)(2ab+2bc+2ca-a^2-b^2-c^2)}{abc}$

Let $a,b,c$ be side lengths of a triangle. What are the minimum and maximum of $$\frac{(a+b+c)(2ab+2bc+2ca-a^2-b^2-c^2)}{abc}?$$ When $a=b=c$, the value is $9$. In addition, we can write $a=x+y,b=y+z,...
0
votes
0answers
17 views

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 "...
0
votes
1answer
20 views

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 ...
10
votes
1answer
124 views

An interesting AM-HM-GM inequality: $\text{AM}+\text{HM}\geq C_n\cdot \text{GM}$

It is not difficult to prove that if $x,y\in\mathbb{R}^+$ the inequality $$ \frac{x+y}{2}+\frac{2}{\frac{1}{x}+\frac{1}{y}}\geq \color{purple}{2}\cdot\sqrt{xy} $$ holds, and the constant $\color{...
2
votes
1answer
52 views

Show that $a+b+c=1$ implies $\exists x, y \in \{a-ab, b-bc, c-ca\}$ so that $x \leq \frac{1}{4}$ and $y \geq \frac{2}{9}$

Let $a, b, c$ be three positive real numbers such that $a + b + c = 1$. Prove that among the three numbers $a − ab, b − bc, c − ca$ there is one which is at most $1/4$ and there is one which is at ...
3
votes
3answers
101 views

Show that $\dfrac{1}{x_{1}(x_{1}+1)}+\dfrac{1}{x_{2}(x_{2}+1)}+\cdots+\dfrac{1}{x_{n}(x_{n}+1)}\ge\dfrac{n}{2}$ with $x_{1}x_{2}\cdots x_{n}=1$

Let $x_{1},x_{2},\cdots,x_{n}$ be postive real numbers, and such $x_{1}x_{2}\cdots x_{n}=1$, Show that $$\dfrac{1}{x_{1}(x_{1}+1)}+\dfrac{1}{x_{2}(x_{2}+1)}+\cdots+\dfrac{1}{x_{n}(x_{n}+1)}\ge\dfrac{...
3
votes
0answers
113 views

Any similar Lagrange's identity inequality

Following problem I have post MO ,we know Lagrange's identity $$(a^2_{1}+a^2_{2}+a^2_{3})(b^2_{1}+b^2_{2}+b^2_{3})=(a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3})^2+\sum_{i=1}^{2}\sum_{j=i+1}^{3}(a_{i}b_{j}-a_{...
2
votes
2answers
49 views

How to verify $(1+\frac{1}{n})^2(1-\frac{1}{n^2})^{n-1}\geq \exp(\frac{1}{n})$

How to verify this inequality? Assuming that $n\in \mathbb{N}^+$, we have: $$\left(1+\frac{1}{n}\right)^2\left(1-\frac{1}{n^2}\right)^{n-1}\geq \exp\left(\frac{1}{n}\right).$$
2
votes
1answer
18 views

Complex numbers inequality $|a_nb_n+…+a_mb_m|≤a_n * \max_{n≤k≤m} |b_n+b_{n+1}+…b_k|$

$a_n,...,a_m \in R $, such that $a_n≥a_{n+1}≥...≥a_m≥0$ and any $b_n,...,b_m \in C$ How to conclude $$|a_nb_n+...+a_mb_m|≤a_n * \max_{n≤k≤m} |b_n+b_{n+1}+...b_k|$$ It is said in the book, that if the ...
3
votes
3answers
120 views

If $a,b>0$ and $a+b=1\;,$ Then minumum value of $(a+\frac{1}{a})^2+(b+\frac{1}{b})^2$ is [duplicate]

If $a,b>0$ and $a+b=1\;,$ Then minumum value of $\displaystyle \left(a+\frac{1}{a}\right)^2+\left(b+\frac{1}{b}\right)^2$ is $\bf{My\; Try::}$ Let $a=\sin^2 \theta$ and $b=\cos^2 \theta\;,$ Then ...
2
votes
1answer
76 views

Find the maximal value of $a+b-c+d$

Let $a, b, c, d$ be real numbers satisfying inequality $$f(x)=a\cos x+b\cos 2x+c\cos 3x+d\cos 4x\le 1$$ holds for $x\in\Bbb{R}$. Find the maximal value of $a+b-c+d$ and determine the values of $a,b,c,...
10
votes
2answers
361 views
+50

Prove $\frac{xy}{5y^3+4}+\frac{yz}{5z^3+4}+\frac{zx}{5x^3+4} \leqslant \frac13$

$x,y,z >0$ 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 $...
2
votes
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 ...
4
votes
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 ...
0
votes
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 ...
0
votes
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 ...
5
votes
1answer
211 views

Prove $\sqrt[2]{x+y}+\sqrt[3]{y+z}+\sqrt[4]{z+x} <4$

$x,y,z \geqslant 0$ 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 ...
17
votes
3answers
527 views

Prove that $\frac{x^x}{x+y}+\frac{y^y}{y+z}+\frac{z^z}{z+x} \geqslant \frac32$

$x,y,z >0$, 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 ...
-2
votes
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!
0
votes
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'...
2
votes
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 $$a^a\cdot b^b+a^b \cdot b^a≤1$$
3
votes
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 $x + iy$ ...
2
votes
1answer
117 views

$(x+2y+z)\cdot \left( \frac{x}{y} +\frac{2y}{z}+\frac{z}{x}\right) > 12$ for $x^2+y^2+z^2=3$

$x,y,z > 0$ 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 ...
4
votes
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 ...
1
vote
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 ...