Questions on proving and manipulating inequalities.

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2
votes
2answers
41 views

Prove inequality $(\sqrt{a} - \sqrt{b})^2 \leq \frac{1}{4}(a-b)(\ln(a)-\ln(b))$

I am trying to prove the following inequality: $$(\sqrt{a} - \sqrt{b})^2 \leq \frac{1}{4}(a-b)(\ln(a)-\ln(b))$$ for all $a>0, b>0$. Does anyone know how to prove it? Thanks a lot in advance! ...
1
vote
0answers
7 views

Solving nonlinear matrix inequality - transformation to LMI

I have a nonlinear matrix inequality problem where $A,B,C$ and $M$ are known and T is unknown and I would like to find $T$ that satisfies $\begin{bmatrix} T^T M T + A & B \\ B^T & ...
1
vote
1answer
29 views

Integral inequality with first two moments equal to $1$.

Let $f\in \mathcal{C}^0([0,1],\mathbb{R})$ such that $$ \int_0^1 f(x)\text{d}x = \int_0^1 xf(x)\text{d}x=1.$$ Show that $\int_0^1 f(x)^2 \ge 4$. I tried to use Cauchy-Schwartz inequality such that ...
2
votes
1answer
24 views

Inequality constant in Papa Rudin

The following inequality appears in Rudin's Real and Complex Analysis, 3ed, in the proof of ($f$) in Theorem 9.2 (Fourier Transforms): if $x\in\mathbb{R}$ and $\phi(x,u):= (e^{-ixu} - 1)/u$ then ...
1
vote
3answers
112 views

Integral Inequality

There is a problem in my book that says: Show that (assuming $f,g,f^2, g^2, $and$ fg$ are integrable) $$\int_{a}^{b}fg\leq \sqrt{\int_{a}^{b}f^2}\sqrt{\int_{a}^{b}g^2}$$. I know this is the ...
3
votes
1answer
36 views

How prove $\binom{n}{m}\le\left(\frac{en}{m}\right)^m$ [duplicate]

Show that $$\binom{n}{m}\le\left(\dfrac{en}{m}\right)^m$$ where $0<m\le n,m,n\in N^{+}$ My idea: since ...
0
votes
3answers
29 views

Is this some kind of triangle inquality?

I stumbled upon the following inequality: $$\Vert x+hz-(x+y)-(p-(x+y))\Vert_2 \geq \Vert p-(x+y)\Vert_2-\Vert x+hz-(x+y)\Vert_2$$ where $p,x,y,z \in \mathbb{R}^n$. My question is: Is this some kind ...
1
vote
2answers
54 views

Odd series convergence

Prove that we have following inequality: $1+ \frac{1}{3} + \frac{1}{5} + ... + \frac{1}{397} > \frac{9}{4}$ Anybody can help me to figure it out?
4
votes
1answer
84 views

determinant inequality, $AB=BA$, then $ \det(A^2+B^2)\ge \det(2AB) $

$A$ and $B$ are two $n\times n $ real matrices, $AB=BA$. Can we conclude that $$ \det \Big(A^2+B^2\Big)\ge \det(2AB) $$ is right? Well, the inequality is interesting. if $A,B$ are upper ...
1
vote
5answers
75 views

Hint in Proving that $n^2\le n!$

The problem is to find the values of n such $n^2\le n!$. I have found this set to be {${n\in Z^+| n\le1 \lor n\ge 4}$} I've used a proof by cases to prove the first part ($\forall n\le 1$) which was ...
0
votes
2answers
63 views

How to solve this inequality?

I have the following problem in an assignment and have been struggling to do it. $2 + 2x - x^2 \geq 2 \sqrt{1+2x}$ I have tried solving for $x$ but have not been able to do so. Any hints to solve ...
1
vote
2answers
30 views

show that inequality holds for real numbers

There are given real numbers $a,b,c,d \in [0,1]$ show that: $a+b+c+d \le 1+a(b+c+d)+b(c+d)+cd$ I tried to transform it to $b(1-a)+c(1-a)+d(1-a)-(1-a)\le b(c+d)+cd$ $(1-a)(b+c+d-1) \le ...
0
votes
1answer
50 views

Cauchy-Schwarz Inequality - Proof using Quadratic Polynomial [Lay P379 Thm 6.7.16]

I don't perceive https://www.dpmms.cam.ac.uk/~wtg10/csineq.html, about why it " is an obvious thing to write down" "a quadratic form, use the fact that it is non-negative everywhere, and ...
4
votes
0answers
64 views

How prove this inequality $(\sum_{k=1}^{3n}a_{k})^3\ge 27n^2\sum_{k=1}^{n}a_{k}a_{n+k}a_{2n+k}$

let $$0\le a_{1}\le a_{2}\le \cdots\le a_{3n}$$ show that $$\left(a_{1}+a_{2}+a_{3}+\cdots+a_{3n-1}+a_{3n}\right)^3\ge 27n^2\sum_{k=1}^{n}a_{k}a_{n+k}a_{2n+k}$$ we know when $n=1$,this is ...
1
vote
2answers
27 views

show that there is not positive integer n satisfying $4n<n^2<5n$

I do not know where to begin, I know that if I choose some n, to be say n=3, this does not hold, and it doesn't hold for any n > 0. But that's not a formal mathematical proof.
0
votes
0answers
23 views

Normal sequences and Montel's Theorem

I am currently stuck on an exam question involving normal sequences and Montel's theorem: Give two examples of non-constant normal sequences one in the $(a)$ unit disk $\mathbb{D}$ and one in $(b)$ ...
4
votes
2answers
78 views

Prove that ax+bx+ay+by ≤ 300.

Let $a,b,x,y$ be positive numbers satisfying: $ax ≤ 100, bx ≤ 100$, $ay ≤ 100, by ≤ 50$. Prove that $ax+bx+ay+by ≤ 300$. Can someone help me ??
0
votes
0answers
27 views

Proving Inequality Equation with $4$ variables [on hold]

How would I approach this question? Let $a, b, x, y$ be positive numbers satisfying: $$a*x ≤ 100,\ b*x ≤ 100,\ a*y ≤ 100,\ b*y ≤ 50$$ Prove that $a*x+b*x+a*y+b*y ≤ 300$.
1
vote
1answer
25 views

Finding the minimum value of a 6th degree polynomial algebraically

Is it possible to answer this question using methods of basic algebra? Find the least value of the expression $a^6 + a^4 - a^3 - a + 1$ for real value of $a$. This question is from the 2013 Philippine ...
1
vote
1answer
38 views

Cauchy-Schwartz Inequality - Proof using Projections [Lay P379 Thm 6.7.16]

If $u=0$, then the inequality becomes $ 0 \le 0 $, which is true. See Practice Problem 6.7.1 on P382. If $u\neq 0$, let $W$ be the subspace spanned by $u$. $1.$ How would one determine to ...
4
votes
2answers
88 views

How find the $AP+\frac{1}{2}BP$ minmum value

An equilateral triangle $ABC$ such $$AB=BC=AC=2a>0$$ A circle $O$ is inscribed in triangle $ABC$,and the point $P$ on the circle $O$. Find the minimum $$AP+\dfrac{1}{2}BP$$ My idea: let ...
7
votes
2answers
85 views

Is $\int_x^{\infty}e^{-\frac{t^2}{2}} < \frac{1}{x}e^{-\frac{x^2}{2}}$?

While solving a problem in real analysis, I got stuck. I need to prove $$\int_x^{\infty}e^{-\frac{t^2}{2}}dt < \frac{1}{x}e^{-\frac{x^2}{2}} $$ Clearly I have to use some kind of inequality, but ...
0
votes
1answer
27 views

$C^{-1} (1+|x|^{2})^{\frac{s}{2}} \leq (1+|x|)^{\frac{s}{2}} \leq C (1+|x|^{2})^{\frac{s}{2}}$?

Let $s\in \mathbb R,$ and define $f: \mathbb R^{n}\to [0, \infty)$ such that $f(x)= (1+|x|^{2})^{\frac{s}{2}}, (x\in \mathbb R^{n})$ and $g:\mathbb R^{n}\to [0, \infty)$ such that $g(x)= ...
7
votes
0answers
33 views

How prove this inequality $\sum_{cyc}\frac{x^a\ln{x}}{(x^a+y+z)^2}\ge 0$

Question: let $x,y,z$ be postive numbers,and such $xyz\ge 1$,and such $a$ is real numbers.show that $$\dfrac{x^a\ln{x}}{(x^a+y+z)^2}+\dfrac{y^a\ln{y}}{(y^a+x+z)^2}+\dfrac{z^a\ln{z}}{( ...
6
votes
4answers
79 views

A question about inequality ${(n+1)\over e^n}^n<n!$

How to prove the inequality $${(n+1)\over e^n}^n<n!$$ I have tried mathematical induction, but it doesn't work! Are there other methods to solve it?
4
votes
1answer
78 views

A tough inequality problem with condition $a+b+c+abc=4$

If, $a+b+c+abc=4$, with $a,b,c$ being positive reals, then prove or disprove the following inequality: $$\frac{a}{\sqrt{b+c}}+\frac{b}{\sqrt{a+c}}+\frac{c}{\sqrt{a+b}}\geq\frac{a+b+c}{\sqrt2}$$ I ...
0
votes
1answer
32 views

Inequality, $\left(\frac{2}{x}+2\right)^{n}-\left(\frac{2}{x}-2\right)^{n}\leq \left(\frac 4 x \right)^n$

How do I show that $$\left(\frac{2}{x}+2\right)^{n}-\left(\frac{2}{x}-2\right)^{n}\leq \left(\frac 4 x \right)^n$$ for $x\in\left(0,1\right]$ and $n\in\mathbb N$?
3
votes
1answer
91 views

counterexamples to $ \det \Big(A^2+B^2\Big)\ge \det(AB-BA) $

$n\geq3$. A and B are two $n\times n$ reals matrices. For $n\times n$, Could one give counterexamples to show that $$ \det \Big(A^2+B^2\Big)\ge \det(AB-BA) \tag{$*$}$$ is not necessarily true? ...
2
votes
4answers
38 views

If $9^{x+1} + (t^2 - 4t - 2)3^x + 1 > 0$, then what values can $t$ take?

If $9^{x+1} + (t^2 - 4t - 2)3^x + 1 > 0$, then what values can $t$ take? This is what I have done: Let $y = 3^x$ $$9^{x+1} + (t^2 - 4t - 2)3^x + 1 > 0$$ $$\implies9y^2 + (t^2 - 4t - 2)y + ...
0
votes
2answers
55 views

Minimum value of $\sqrt{(1+1/y)(1+1/z)}$

If $y,z > 0$ and $y + z = c$ where $c$ is a constant, then what's the minimum value of $$\sqrt{\left(1+\frac1y\right)\left(1+\frac1z\right)}$$ I am having a hard time solving this.
7
votes
1answer
87 views

How to learn inequalities and become good at proving them?

I am taking a real analysis course next year and I want to start slowly preparing for that class now, so I hope you can help me. The class is quite challenging and the fail rate is relatively high. ...
1
vote
3answers
45 views

Quadratic equations and inequalites

For every positive integer $n$, prove that $$\sqrt{4n+1}<\sqrt{n} + \sqrt{n+1}<\sqrt{4n+2}$$ Hence or otherwise, prove that $[\sqrt{n}+\sqrt{n+1}] = [\sqrt{4n+1}]$, where $[x]$ ...
0
votes
1answer
17 views

Nonexpansive Affine Operators in Hilbert spaces

Let $H$ be a Hilbert space with real inner product $\langle \cdot, \cdot \rangle : H \times H \rightarrow \mathbb{R}$. Let $T$ be an affine operator. Show that $T$ is nonexpansive, i.e., $\left\| ...
1
vote
0answers
43 views

How prove this $\sin(2a)+\sin(2b)+\sin(2c)<\dfrac{\pi}{2}+2\sin a\cos b+2\sin b\cos c$

let $0<a<b<c<\dfrac{\pi}{2}$, use the integral inequality show that $$\sin(2a)+\sin(2b)+\sin(2c)<\dfrac{\pi}{2}+2\sin a\cos b+2\sin b\cos c$$ I know this problem can use The area of ...
1
vote
3answers
64 views

problem with inequality of modulus

how can I prove the following inequality? $$\frac{\mid x+y\mid}{1+\mid x+y\mid}\leq\frac{\mid x \mid}{1+\mid x\mid}+\frac{\mid y \mid}{1+\mid y \mid}$$ I was trying to prove it by ...
5
votes
1answer
71 views

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

For positive real numbers with $a+b+c=abc$ prove that $$\frac{1}{\sqrt{1+a^2}}+\frac{1}{\sqrt{1+b^2}}+\frac{1}{\sqrt{1+c^2}}\le\frac{3}{2}$$ I made the substitution $a=\tan(\alpha), b = ...
5
votes
4answers
124 views

For all reals $x$, prove $2^x > x$

How can I prove that for all reals $x$, $2^x > x$? I can prove this for integers with induction, but I can't figure out how to prove it for reals. Perhaps you could say that since $2^x$ is strictly ...
0
votes
1answer
19 views

Norm and Matrice Proof

I'm trying to show that the following statement is true: If $||\mathbf a - \theta \mathbf b||^2 - ||\mathbf a||^2 \geq 0$ for all $\theta \in [0,1]$, then $\mathbf a^T \mathbf b \le 0$. Is this ...
2
votes
1answer
120 views
+50

How prove this $|S_{1}|-|S_{2}|\le 2^{2n}\binom{2n}{n}$

Question: Let $n\in \mathbf N^{+}$,and define set $S=\{1,2,\cdots,4n\}$, for any $ a<b\in \mathbf R^{+}$, define $$S_{1}=\{\,X\mid X\subseteq S,a\le S(X)\le b,S(X)\equiv 1\pmod 2\,\}$$ ...
-4
votes
0answers
22 views

Quadratic Equation Inequality [closed]

a,b,c,p,q,r are real numbers such that ax^2+bx+c>=0,px^2+qx+r>=0 For all real numbers prove that apx^2+bqx+cr>=0
1
vote
0answers
32 views

Prove or disprove $\sum_{j=1}^{n}\sum_{i=1}^{n}|a_{i}-a_{j}|^2|b_{ij}|^2\le \cdots$

let $a_{i},b_{ij}\in C,i=1,2,\cdots,n,j=1,2,\cdots,n$,prove or disprove $$\sum_{j=1}^{n}\sum_{i=1}^{n}|a_{i}-a_{j}|^2|b_{ij}|^2\le ...
2
votes
3answers
43 views

Proof by induction of a recursive sequence

I am studying CIE A levels Further Maths and I am stuck at a question from June 2002: Q The sequence of positive numbers $u_1,u_2,u_3,...$ is such that $u_1<4$ and ...
2
votes
0answers
26 views

Inequality similar to Hoeffding

I have a coin with heads probability $p_1$. I toss it $n_1$ times. Let $\hat{p}_1$ be the empirical heads probability. Then we know from Hoeffding that $$P\left( \left|\hat{p}_1-p_1 \right| \geq ...
0
votes
2answers
22 views

Inequality: $\tan(x) > 1$

So far, I've not come very... far. It ends up with me trying to solve it more intuitively than mathematically. I figured, first I'll find the place of equality, which is at $x = \arctan 1 = ...
2
votes
4answers
46 views

If $a^2+b^2=1$ where $a,b>0$ then find the minimum value of $(a+b+{1\over{ab}})$

If $a^2+b^2=1$ where $a,b>0$ then find the minimum value of $(a+b+{1\over{ab}})$ This can be easily done by calculas but is there any way to do do this by algebra
1
vote
1answer
12 views

Determine the values of x for which the linear approximation is accurate to within 0.1.

So I've got a function: $$\frac{1}{(1+2x)^4}$$ with its linear approximation: $$1-8x$$ For all values of $x$ where the linear approximation is accurate within $0.1$, then surely we subtract the ...
5
votes
1answer
141 views

determinant inequality $ \det(A^2+B^2+(A-B)^2)\ge 3\det(AB-BA) $

A and B are two $2\times2$ reals matrices. then $$ \det \Big(A^2+B^2+(A-B)^2\Big)\ge 3\det(AB-BA) $$ well, it is seems interesting, but it is really hard to get started Thank you very much!
0
votes
0answers
44 views

what is the difference between $\log_ax^2$ and $2\log_ax$

when are we allowed to make the use of the formular $\log_ab^c=c\log_ab$ for example solving the logarithmic inequality $\log _2(-t)+\log_2(t)^2<3$ is solveable if $t <0$ but $\log ...
0
votes
1answer
34 views

Solve a system of inequalities

$$\begin{cases} \log_{2}^{2}(-\log_{2}x) + \log_{2}\log_{2}^{2}x \leq 3 & \\-4 |x^2-1|-3\geq \frac{1}{x^2-1}& \end{cases}$$ What I've tried: Make substitution $t=x^2-1$ and solve second ...
5
votes
7answers
115 views

Find value range of $2^x+2^y$

Assume $x,y \in \Bbb{R}$ satisfy $$4^x+4^y = 2^{x+1} + 2^{y+1}$$, Find the value range of $$2^x+2^y$$ I know $x=y=1$ is a solution of $4^x+4^y = 2^{x+1} + 2^{y+1}$ , but I can't go further more. I ...