Questions on proving, manipulating and applying inequalities.

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1
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2answers
71 views

Dominance between two functions

Let two functions $f(z)$ and $g(z)$ with $z\in[0,c]$ with $c$ a constant such that $c<b$. I'd like to check whether $f(z)-g(z)>0$. I've tried to set $f(z)$ to its minimal value and $g(z)$ to its ...
5
votes
4answers
135 views

Prove that $10^{340} < \dfrac{5^{496}}{1985}$

Prove that $10^{340} < \dfrac{5^{496}}{1985}$. I said since $2^{13} < 10^{4}$, we see that $5 = \dfrac{10}{2} > 10^{\frac{9}{13}}$ and so $10^{340} < \dfrac{10^{343.38}}{1985} <\dfrac{...
-10
votes
2answers
50 views

Which quantity is greater? [on hold]

A. -0.1 or B. -0.10101010101 This is actually an evaluation of an expression when plugging certain values in the GRE. I plugged in the value -0.1 and arrived at my doubt
0
votes
1answer
41 views

a inequality similar to geometric means

Let $a$, $b$ be two positive constants. We sure have $$ a^2+b^2\geq 2ab $$ My question: would it be possible to have an inequality like $$ a^2+b^2\geq Ca^{2+\epsilon}b^{1-\eta} $$ where $C$, $\epsilon$...
5
votes
3answers
139 views

Prove the Inequality $\frac{1}{1-x}-\frac{x(3-x)(2-x)(13x^4-50x^3+89x^2-84x+36)}{4(1-x)(2x(1-x))^2}<1$

Can anyone suggest any hints to prove the following inequality: $$\frac{1}{1-x} - \frac{x(3-x)(2-x)(13x^4 - 50x^3 + 89x^2 - 84x + 36)}{4(1-x)(2x(1-x))^2} < 1,$$ for all $x \in (0,1)$?
0
votes
1answer
69 views

Find maximum value of a function [closed]

$a$, $b$, and $c$ are real numbers, and $a+b+c=0$ and $a^2+b^2+c^2=2$. I need help finding the maximum value of: $$\big|a^2b^2(a-b)+b^2c^2(b-c)+c^2a^2(c-a)\big|$$ To be honest, I don't know where ...
4
votes
1answer
204 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 ...
1
vote
2answers
66 views

Inequality involving sum of logarithms and hidden zeta-function

I would like to prove the following estimation: if $n \ge 2$ is a natural number, then $$\sum_{k=2}^n \frac{\log^2 k}{k^2} <2 - \frac{\log^2 n}{n}.$$ I have noticed that LHS is indeed bounded by ...
17
votes
3answers
689 views

Prove $(x+y)(y^2+z^2)(z^3+x^3) < \frac92$ for $x+y+z=2$

$x,y,z \geqslant 0$ and $x+y+z=2$, Prove $$(x+y)(y^2+z^2)(z^3+x^3) < \frac92$$ While numerical method can solve this problem, I am more interested in classical solutions. I tried this problem for ...
0
votes
1answer
43 views

Comparing the roots of two increasing functions

For any $0 \leq x \leq y \leq 1$, define $f(y;x):=\frac{y^2}{2}-\frac{2 y^3}{3}+\frac{y^4}{4} - \frac{x^2}{2} + \frac{x^3}{3}$ and $g(y;x):=\frac{y^2}{3}-\frac{2 y^3}{4}+\frac{y^4}{5} - \frac{x^2}{3} +...
0
votes
0answers
28 views

Another inequality from $(x_1, . . . , x_n)$ majorizes $(y_1, . . . , y_n)$

I am looking for have a proof of my problem as following: Let $I$ be an interval of the real line and let $f$ denote a real-valued, convex function defined on $I$, and $f' \geq 0$ on $I$. If $x_1, . ....
0
votes
1answer
38 views

Using CS inequality to find maximum of a function

I am trying to us Cauchy-Schwarz inequality to find the maximum of: $$|(a^2)(b^2)(a-b)+(b^2)(c^2)(b-c)+(c^2)(a^2)(c-a)|$$ Where $a$, $b$, and $c$ are real numbers, and $a+b+c=0$ and $a^2+b^2+c^2=2$. ...
-1
votes
0answers
31 views

Solve Equation with max integer [closed]

Solve please $\dfrac{\left[\sqrt{x-[x ]}\right]}{(x+3)(x+4)}\ \geq0$ edit
1
vote
3answers
41 views

Show $d_2(f,g) = \sqrt{\int\limits_0^1 |f(x) - g(x)|^2 dx}$ is a metric on $C[0,1]$

I am surprised that this question hasn't been asked on here I need to show that $$d_2(f,g) = \sqrt{\int\limits_0^1 |f(x) - g(x)|^2 dx}$$ is a metric on $C[0,1]$ Proof: As usual, positive ...
0
votes
1answer
31 views

Solving inequality of two independent exponentially distributed RVs

I have huge problems solving following excersice: There are two molecules. The decay of the molecules is exponentially distributed with $\alpha_1 = 1$ (for molecule 1) and $\alpha_2 = 2$ (for ...
1
vote
1answer
75 views

Square root inequality revisited

This is a follow-up question of this one: Proof of the square root inequality I am interested in the following generalizations of the square root inequality. Let $\varepsilon,\delta>0.$ Then $$\...
0
votes
2answers
45 views

Determining values satisfying an inequality

I have the following inequality: $$\left\lceil \frac{\log((n-1)/6000)}{\log(3)} \right\rceil < \left\lceil \frac{\log((n-1)/3000)}{\log(3)} \right\rceil,$$ where $n$ is a positive integer, and I ...
2
votes
1answer
31 views

If $N = q^k n^2$ is an odd perfect number with Euler prime $q$, and $k=1$, does it follow that $\frac{\sigma(n^2)}{n^2} \geq 2 - \frac{5}{3q}$?

Let $\sigma=\sigma_{1}$ be the classical sum-of-divisors function. If $N = q^k n^2$ is an odd perfect number with Euler prime $q$ (i.e., $q$ satisfies $q \equiv k \equiv 1 \pmod 4$), and $k=1$, does ...
1
vote
1answer
30 views

When is the following inequality true? $ \int_0^1 \lvert f-g\rvert \leqslant \max_{x\in[0,1]}\lvert f(x)-g(x)\rvert $

Let $ f,g \in \mathcal{C}([0,1])$. Then we have $$ \int_0^1 \lvert f(x)-g(x)\rvert dx \leqslant (1-0)\max_{x\in[0,1]}\lvert f(x)-g(x)\rvert $$ Now, is it only true for continuous functions in $[0,...
7
votes
4answers
148 views

inequality $\sqrt{\cos x}>\cos(\sin x)$ for $x\in(0,\frac{\pi}{4})$

How can I prove the inequality $\sqrt{\cos x}>\cos(\sin x)$ for $x\in(0,\frac{\pi}{4})$ ? The derivative of $f(x):=\sqrt{\cos x}-\cos(\sin x)$ is very unpleasant, so the standard method is ...
3
votes
2answers
46 views

$3(x-\sqrt{xy}+y)^2\geq x^2+xy+y^2$

Prove that $3(x-\sqrt{xy}+y)^2\geq x^2+xy+y^2$ for all $x,y\geq 0$. Expanding, the inequality becomes $$3x^2+3xy+3y^2-6x\sqrt{xy}-6y\sqrt{xy}+6xy\geq x^2+xy+y^2$$ which is $$x^2+4xy+y^2\geq3\sqrt{xy}(...
1
vote
0answers
32 views

Checking feasibility of a system of inequalities with scipy

I have a set of pairwise constraints, like this: a > b, b > c, c > a and need to check if they are satisfiable (in the example above, they are not). ...
1
vote
2answers
54 views

Proof that $\sum\limits_{k=1}^\infty\frac{2k}{(k^2+c^2)^2}\gt\frac{2}{2c^2+1}$

I tried to prove the following inequality which gives a lower bound to the Mathieu sum: $$S=\sum_{k=1}^\infty\dfrac{2k}{(k^2+c^2)^2}$$ where $c\neq0$. The Mathieu inequality states: $S\lt\dfrac{1}{c^2}...
1
vote
1answer
36 views

Positivity of a quartic form

If I have a quartic form that I can write as $$P(x,y)=(x^2/2,y^2/2,xy)M(x^2/2,y^2/2,xy)$$ where $M$ a a $n \times n$ symmetric matrix, what is the simplest way to derive whether the form is positive ...
0
votes
1answer
41 views

Solution for an inequality

I want to solve this inequality for $z$ $$(z+1) \left(1-e^x\right)-e^y\geq 0$$ where $-\infty <x\leq \log \left(\frac{1}{z+1}\right)$ and $-\infty <y\leq 0$. I am struggling because $z$ ...
7
votes
3answers
6k views

Properties of $\liminf$ and $\limsup$ of sum of sequences: $\limsup s_n + \liminf t_n \leq \limsup (s_n + t_n) \leq \limsup s_n + \limsup t_n$

Let $\{s_n\}$ and $\{t_n\}$ be sequences. I've noticed this inequality in a few analysis textbooks that I have come across, so I've started to think this can't be a typo: $\limsup\limits_{n \...
1
vote
0answers
32 views

Showing an inequality of LimSup: $\limsup(x_n+y_n)\geq\limsup(x_n)+\liminf(y_n)$ [duplicate]

Let $\{x_n\}_{n=1}^\infty$ and $\{y_n\}_{n=1}^\infty$ be sequences of real numbers. Verify that each of the following holds, provided the right-hand side makes sense. $$\limsup(x_n+y_n)\geq\limsup(x_n)...
0
votes
1answer
66 views

On the inequality $|z_1-z_2|^2 \lt (1+c)|z_1|^2+(1+\frac{1}{c})|z_2|^2$

Now, I know this question has been asked here but my question doesn't deal with finding a solution, my question deals with checking the validity of the question. Question:- If $z_1, z_2$ are ...
1
vote
4answers
1k views

Average of square roots's sum vs. square root of an average

I was watching a video on youtube about how colors work in computers, and found this statement: "The average of two square roots is less than the square root of an average" The link to the ...
5
votes
1answer
52 views

Bound on $c-b$ for $a^n+b^n=c^n$

Let $a\leq b\leq c$ be positive real numbers and $n$ positive integer with $a^n+b^n=c^n$. Prove that $c-b\leq(\sqrt[n]{2}-1)a$. The desired inequality can be written as $c-b+a\leq \sqrt[n]{2}a$. ...
7
votes
1answer
196 views

An Inequality Involving The Riemann Zeta Function

I'm having trouble proving the following inequality for $2<r<3$: $$(1+2^{-r})\frac{(3^r+1)^2}{3^{2r}+1}>\frac{\zeta(r)}{\zeta(2r)}.$$ I can easily plot the graph, and the inequality clearly ...
-2
votes
0answers
51 views

To prove that roots of a polynomial are real. [closed]

Given three polynomials $$ f(x)=x^2+a_1x+b_1, \\ g(x)=x^2+a_2x+b_2, \\ h(x)=x^2+a_3x+b_3 $$ such that $$a_1a_2a_3=b_1b_2b_3>1.$$ Prove that one of these polynomials has 2 distinct real roots.
1
vote
1answer
27 views

Norm of multiplication and multiplication of norms

It is well known that $\|u \cdot v \|_2 \le \|u \|_2 \cdot \| v \|_2 $ for all $u, v \in \mathbb{R}^d$. Is the following true for all $p \in \mathbb{R}$: $$\|u \cdot v \|_p \le \|u \|_p \cdot \|...
0
votes
2answers
35 views

Clarification on inductive proof of Bernoulli's inequality

Prove that if $h > -1$, then $1 + nh ≤ (1+h^n)$ for all nonnegative integers $n$. I've read several solutions and I'm still totally lost on how to go about this. I have the inductive hypothesis:...
0
votes
1answer
40 views

Inequality with two absolute value

How can you tackle an inequality problem that has two absolute values? Example is the following $p + |k| > |p| + k$ and the questions is a quantitative comparison between A) $p $ B) $k$ The ...
8
votes
3answers
496 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 ...
3
votes
2answers
151 views

Inequality and Induction: $\prod_{i=1}^n\frac{2i-1}{2i}$ $<$ $\frac{1}{\sqrt{2n+1}}$ [duplicate]

I needed to prove that $\prod_{i=1}^n\frac{2i-1}{2i}$ $<$ $\frac{1}{\sqrt{2n+1}}$, $\forall n \geq 1$ . I've atempted by induction. I proved the case for $n=1$ and assumed it holds ...
1
vote
2answers
152 views

Mathematics induction on inequality: $2^n \ge 3n^2 +5$ for $n\ge8$

I want to prove $2^n \ge 3n^2 +5$--call this statement $S(n)$--for $n\ge8$ Basis step with $n = 8$, which $\text{LHS} \ge \text{RHS}$, and $S(8)$ is true. Then I proceed to inductive step by ...
1
vote
7answers
125 views

Induction and convergence of an inequality: $\frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots(2n)}\leq \frac{1}{\sqrt{2n+1}}$

Problem statement: Prove that $\frac{1*3*5*...*(2n-1)}{2*4*6*...(2n)}\leq \frac{1}{\sqrt{2n+1}}$ and that there exists a limit when $n \to \infty $. , $n\in \mathbb{N}$ My progress LHS is ...
1
vote
3answers
107 views

Proof of an inequality by induction: $(1 + x_1)(1 + x_2)…(1 + x_n) \ge 1 + x_1 + x_2 + … + x_n$

Let $n \in \mathbb N^+$. Show that if $x_1, x_2, ... , x_n$ are $n$ real numbers such that $-1 \le x_i \le 0$ for each $1 \le i \le n$, then $$(1 + x_1)(1 + x_2)...(1 + x_n) \ge 1 + x_1 + x_2 + ... + ...
2
votes
3answers
53 views

Proof by induction: inequality $n! > n^3$ for $n > 5$

I'm given a inequality as such: $n! > n^3$ Where n > 5, I've done this so far: BC: n = 6, 6! > 720 (Works) IH: let n = k, we have that: $k! > k^3$ IS: try n = k+1, (I'm told to only work ...
4
votes
4answers
136 views

Prove by mathematical induction that: $\forall n \in \mathbb{N}: 3^{n} > n^{3}$

Prove by mathematical induction that: $$\forall n \in \mathbb{N}: 3^{n} > n^{3}$$ Step 1: Show that the statement is true for $n = 1$: $$3^{1} > 1^{3} \Rightarrow 3 > 1$$ Step 2: Show ...
-2
votes
3answers
86 views

Prove the inequality by induction: $3^n > n^3$ for $n\ge4$ [duplicate]

Prove the inequality by induction: $3^n > n^3\ $ for $\ n \geq 4$ Edit: 1) Base case: $n=4$, $3^4>4^3, 81>64$ 2) Assume true for n=k: so $3^k>k^3$ 3) Consider $(k+1)^3$, $(k+1)^3 = k^...
1
vote
2answers
108 views

Proof by induction; inequality $1\cdot3+2\cdot4+3\cdot5+\dots+n(n+2) \ge \frac{n^3+5n}3$

Ok so I'm kind of struggling with this: The question is: "Use mathematical induction to prove that 1*3 + 2*4 + 3*5 + ··· + n(n + 2) ≥ (1/3)(n^3 + 5n) for n≥1" Okay, so P(1) is true as 1(1+2)=3 and (...
1
vote
2answers
120 views

Trying to prove $2( \sqrt{n+1}-\sqrt n )< \frac{1}{\sqrt n}<2( \sqrt{n}-\sqrt {n-1})$ and use this to prove… [duplicate]

I am trying to prove this $2( \sqrt{n+1}-\sqrt n )< \frac{1}{\sqrt n}<2( \sqrt{n}-\sqrt {n-1})$ if $n \ge 1$ and using this to prove $2\sqrt{m}-2<\sum^m_{n=1} \frac{1}{\sqrt n}<2( 2\sqrt{m}...
3
votes
0answers
47 views

Linear separability / Number of positive solutions of a random linear system

This one is on linear separability of cyclic patterns. The shorter geometric version: Take a ring of length $p$ of randomly assigned mean-free binary values $x_i = \pm 1$, $i = 1 \cdots p$. ...
1
vote
1answer
40 views

how to find the solution for this inequality?

The question is $(2+\sqrt3)^{x^2-x}+(2-\sqrt3)^{x^2-x}\ge14$ how will i proceed with this question? I'm not able to think of any idea of how to solve this question please help with this question
0
votes
1answer
28 views

if $\sqrt{\log_4(\log_3(\log_2(x^2-2x+a)))}$ is defined for all x={set of real numbers} then find the valid interval for “a”.

if $\sqrt{\log_4(\log_3(\log_2(x^2-2x+a)))}$ the question asks to find the interval in which the valid values for "a" lies I tried by defining the ${\log_4(\log_3(\log_2(x^2-2x+a)))}$$\ge$0
2
votes
0answers
24 views

On some iterated inequalities and $x \geq 5$

Let $x \in \mathbb{N}$. Suppose that I have a function $f:\mathbb{N}\rightarrow\mathbb{Q}$, with initial bounds $$2 - \frac{2}{x_0} < f(x_0) = \frac{2{x_0}}{x_0 + 1} \leq 2 - \frac{5}{3x_0}.$$ ...
1
vote
1answer
34 views

Existence of a functional inequality

Does there exist $f=f(x)$ satisfying $f(x)\ge0$ for $x\in\mathbb{R}$, $f(x)=f(-x)$ for $x\in\mathbb{R}$ (i.e. $f$ is even), $\int_{\mathbb{R}} f(x)\,dx<\infty$, and $\int_{\mathbb{R}} x^2\,f(x)\,dx&...