# Tagged Questions

For questions about proving and manipulating functional inequalities.

18 views

88 views

### Inequalities help! [closed]

Let $a,b,c > 0$ and $a + b + c = 1$. Prove: $$\sqrt{\frac{ab}{c + ab}} + \sqrt{\frac{bc}{a + bc}} + \sqrt{\frac{ac}{b + ac}}\leq \frac32$$
74 views

### When does $f\left [g\left (x \right ) \right ]=g\left [ f\left ( x \right ) \right ]$ have roots.

I met a interesting equation: $$\sin\left [\cos\left (x \right ) \right ]=\cos\left [\sin\left (x \right ) \right ]$$ (And of course, the equation has no roots). So, Let $f\left ( x \right )$ ...
167 views

### Proving an inequality involving discrete variables

I'm trying to show that the following inequality holds $$\frac{1-x^{n}}{1-x^{n+1}}\geq\frac{\sum_{i=0}^{n-2}x^{i}(1-x_{1}^{n-(i+1)})}{\sum_{i=0}^{n-1}x^{i}(1-x_{1}^{n-i})},$$ where $n$ is a ...
73 views

### Need help in solving inequality of this type: $|ax + b| > -c$

This is the equation: $|3x+6|>-12$. I solved it under two cases ($3x+6>-12$) and ($3x+6<12$). More than the answer, what I need to know is - Have I rightly constructed those $2$ cases, if ...
59 views

### If $\|f\|_X\le c_1\|f\|_Y$ then do we have $\langle f,g\rangle_X\le c_2\langle f,g\rangle_Y$?

Let $X$ and $Y$ be two inner product spaces with inner products $\langle \cdot,\cdot\rangle_X$ and $\langle \cdot,\cdot\rangle_Y$, respectively. Suppose we have $\|f\|_X\le c_1\|f\|_Y$ for any $f\in Y$...
49 views

### Suppose that $g(x) = \frac{f(x+1)-f(1)}{f(x)-f(0)} \geq g(1)$, what do we know about $f$?

Suppose $f:\mathbb{R_+} \to \mathbb{R}$ is a continuous and strictly increasing function. Define $g(x) = \frac{f(x+1)-f(1)}{f(x)-f(0)}$. For which $f$ the function $g$ satisfies $g(x) \geq g(1)$ for ...
114 views

### How to show $\Big\vert \frac{\sin(x)}{x} \Big\vert$ is bounded by $1$?

This may be a silly question, but I cannot figure it out. I want to prove that $\Big\vert \frac{\sin(x)}{x} \Big\vert \leq 1$ for $x\in[-1,0)\cup(0,1]$, but I don't even know where to start.
23 views

### Condition on the limit of the upper incomplete gamma function

I am trying to find a lower bound on $q$ such that $$\Gamma(2p,qt)=\int_{qt}^{\infty}{x^{2p-1}e^{-x}\,dx}>0$$ for all $t>0,p>0$, and $q<0$. At first, I tried expanding $\Gamma(2p,qt)$ ...
1k views

Problem: Find all $x$ such that $|x^2-3x+1|<1$ I can't understand how to get started with this. I've never tried to solve quadratic Inequalities before. At first I thought of working with the ...
64 views

### Doubt with Absolute Value Inequality

Problem: Find all values of $x$ for which $\dfrac{|x-2|}{x-2}>0$ My incorrect attempt: Using the definition the Modulus, $|x-2|=x-2$ for all $x\ge2$ and $|x-2|=-x+2$ for all $x\le2.$ ...
54 views

### Prove that a functional is convex

Let $T$ be a self-adjoint bounded operator on a Hilbert space $H$. We define the functional $\Phi$ as: $$\Phi(x)=\frac{1}{2}(Tx,x)$$ My exercise says that $\Phi$ is ...
34 views

### Bounds on functions via its derivatives

Suppose, we have a function $f$ where $f$ is: Contionuos. Non-Negative Has a derivative given by $f'$. Can we have a bound on $f$ in terms of its derivative $f'$? That is have an inequality that ...
77 views

49 views

### Poincaré type inequality

Consider a sequence of functions $f_n \in H^1(M)$, the first Sobolev space of a complete (possibly noncompact) Riemannian manifold. If we have the normalization $\Vert \nabla f_n\Vert_{L^2} = 1$, ...
21 views

### Understanding the Derivation of Dual Geometric Programming Problem

Enthusiastic CS major interested in Optimization Theory here. Pardon me for overlooking something obvious. I'm referring to this nice tutorial/ebook: http://faculty.uml.edu/cbyrne/optfirst0.pdf In ...
71 views

### A question on (odd) perfect numbers

(Note: This has been cross-posted to MO.) Let $\sigma(x)$ be the (classical) sum of the divisors of $x$. A number $N \in \mathbb{N}$ is called perfect if $\sigma(N)=2N$. An even perfect number $U$ ...
2k views

### Inequality involving Square Root

$\sqrt{3x-x^2}<4-x$ I know I can't simply square both sides of an inequality. I have narrowed down the possible values of x => x belongs to [0,3] because the expression inside the square root ...
33 views

### help me to prove this

I would like to prove that $n/k!$ for $k$ that holds $k\leq \frac{\log n}{\log\log n}$ will always be bigger/equal to $1/2$. I tried to use stirling but got stuck. Any ideas? Thanks, Jonatan
66 views

### Is $x(x!)^{1/x}$ an increasing function of $x$, for $x > 0$?

Is $x(x!)^{1/x}$ an increasing function of $x$, for $x > 0$? Here $x!$ is the factorial of $x$. Sure, I do know differential calculus, but my problem is that I do not know how to compute for the ...
129 views

### Numerical methods to solve nonlinear system of inequalities?

I know some methods to solve nonlinear system of equaltites: Relaxation Method, Newton method, nonlinear Jacobi method, nonlinear Seidel method. Is it exist some analogous method to solve nonlinear ...
61 views

### E(X)*E(1/X) <= $(a+b)^2$/4ab

I've worked on the following problem and have a solution (included below), but I would like to know if there are any other solutions to this problem, particularly more elegant solutions that apply ...
85 views

144 views

71 views

59 views

### Trigonometric inequality $|\sin{a_1}|+|\sin{a_2}|+…+|\sin{a_n}|+|\cos{(a_1+a_2+…+a_n)}| \ge1$ for all real $a_i$

Prove that for all real numbers $a_1,a_2,...,a_n$ the following inequality holds: $$|\sin{a_1}|+|\sin{a_2}|+...+|\sin{a_n}|+|\cos{(a_1+a_2+...+a_n)}| \ge 1$$
40 views

56 views

### Inequality to bound $\sum_i a_i b_i - \sum_i c_i d_i$ (harmonic eigenfunction/graph) type sum with constraints

We have a homogeneous graph $G = (V,E)$ with a function $f:V\rightarrow \mathbb{R}$. We define the following modulus: $\displaystyle \omega(s) = \sup\{f(x)-f(y) \ | \ |x-y|=s \}$ and wish to lower ...
70 views

173 views

### Difficult but Interesting Inequalities Problems

1.) Consider the identity $$(px + (1-p)y)^2 = Ax^2 + Bxy + Cy^2.$$ Find the minimum of $\max(A,B,C)$ over $0 \leq p \leq 1$. 2.) Let $n$ be a positive integer. Show that the smallest integer ...
538 views

### Does any one-to-one function exist that satisfies this inequality for all real numbers?

Does there exist a one-to-one function $f: \Bbb R \to \Bbb R$ such that $f(x^2) - (f(x))^2 \geq \frac 1 4\ \ \forall x \in \Bbb R$ ? I've tested this with many one-to-one functions but the ...
Suppose that I have a (generic) function $g$ and arguments $a, b \in \mathbb{N}$. I know that $g$ satisfies the inequalities $$1 < \frac{g(b)}{b} < \frac{g(a)}{a} < 2.$$ I also know that ...