Questions tagged [functional-inequalities]

For questions about proving and manipulating functional inequalities.

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3answers
90 views

Finding range of values of trigonometric function

$f(n)= (\sin x)^n + (\cos x)^n$ I wanted to find the range of the values of this function in terms of n . What I tried I tried to use various inequalities like AM, GM ,HM but failed to derive ...
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0answers
79 views

Energy estimate

Let $V \in \mathcal{C}^{\infty} ([0,T] \times [-R,R])$ such that : $$\partial _t^2 V - cV = F,$$ for some constant $c >0$ and some $F \in L^2( [0,T] \times [-R,R])$, with initial conditions $V(0,x)=...
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0answers
68 views

How to ensure that sum of exp function always positive?

I have a model which looks like $$ f(\mathbf{x}) = \sum_{i=1}^N\lambda_i\exp\left[-\frac{1}{2}(\mathbf{x} - \mathbf{\mu}_i)^T \Sigma^{-1} (\mathbf{x} - \mathbf{\mu}_i)\right] $$ The $\Sigma$ is ...
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1answer
97 views

Information inequality

Information inequality If $\theta_0$ is identified $[\theta \neq \theta_0,\implies f(z, \theta) \neq f(z, \theta_0)]$ and $E [\ln f(z, \theta) ] < \infty$ for all $\theta$ then $L(\theta) = E[\ln f(...
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0answers
37 views

Proof of the inequality $(2n/\textrm{log}n -1)/\sqrt{n} \geq \textrm{log} n$ for sufficiently large n?

I found this inequality in CLRS (Coreman)(Introduction to Algorithms 3rd Edition), on Page - 137 . I can't understand the idea behind the proof.It would be really helpful if someone could provide the ...
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2answers
25 views

Is 1 point homogeinety (essentially) , with F(1)=1; F:[0,1]\to[0,1], equivalent to, Cauchy's equation and continuity, by itself?

It is said that a function is linear if if satisfies Cauchy's equation (A): (A)$$\forall(x,y)\,\in\,\mathbb{R};\quad F(x +y)=F(x)+F(y)$$ And 1 Point Homogeneity $(B)$: (B)$$ \forall \sigma , x\, \...
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0answers
86 views

Is this true? A strict monotonic function F satisfying Jensen equation with F(0)=0, only expresses dyadic rationals multiples?

. Is is really true that a function satisfying jensen equation F(x/2+y/2)=F(x)/2+F(y)/2, F(0)=0, and F strictly monotonic, where F is a function from F:[0,1] to [0,1] can only express dyadic fraction ...
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1answer
129 views

Strictly Monotonic increasing, Symmetric ,doubling Function $F:[0,1]\to [0,1]$

*strong text****Is this symmetric, doubling,**strictly monotone increasing function $F:[0,1]\to[0,1]$ below, continuous and $F(x)=x$ over the unit interval? Let $F: [0,1]\to [0,1],F(1)=1,F(\frac{1}{...
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0answers
93 views

How does one prove, or what is required for a 'strictly monotonically increasing function'F,(1)to be continuous and (2)/or surjective

What is generally required to ensure that a strictly monotonically increasing function,is continuous and/or surjective? F is an (injective) function F: $[0,1]\to [0,1]$ 2. F strictly monotonically ...
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2answers
910 views

two variables quadratic inequalities solution

Suppose there are $n$ quadratic inequalities, the form is $A_i x^2 + B_i y^2 + C_i xy + D_i x + E_i y + F_i \leq 0$, $(\forall i \in [1,n])$, where $x,y$ are two variables and $(A_i, B_i, C_i, D_i, ...
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1answer
38 views

How does $af\left(\frac{n}{b}\right) \leq cf(n)$ imply that $a^{i}f\left(\frac{n}{b^{i}}\right) \leq c^{i}f(n)$?

This is part of a proof for the third case in the Master Theorem in [CLRS], 3rd edition. $a\geq 1$, $b>1$ and $c<1$. Also, $f$ is a nonnegative function. It makes sense for polynomial ...
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1answer
83 views

Is decimal part of inequality whole number?

If you have an inequality, x is "greater than -7" but "less than -5." Is -6 the only number that will satisfy this inequality, or will there be multiple solutions (e.g. -5.5, -6.5, etc)? In other ...
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0answers
62 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 $f\in Y$...
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0answers
49 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 $\Gamma(2p,qt)$ ...
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1answer
61 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 $\Vert \nabla f_n\Vert_{L^2} = 1$, ...
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1answer
92 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 ...
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1answer
60 views

The functional inequality $f(|x|)+f(|y|) \geq 1/f(|x+y|)$

Please help me to solve the following problem. Does there exist a nonempty function $f: D_{f} \subset \mathbb{R} \to \mathbb{R}$ with $D_{f} \neq \emptyset$ such that $$ f(|x|)+f(|y|) \geq \frac{1}{f(...
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1answer
71 views

Is it correct to approach this with Holder Inequality? What am I doing wrong?

I know that $ \forall n\in N, a_0 + a_1 + ... + a_n = 1$, with $a_0, a_1, ... a_n > 0$ and $f(t) = a_0t^n+a_1t^{n-1}+...+a_n, \forall t\in R$ I have to prove that for every $x > 0$ $$ f^2\left(...
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1answer
69 views

Find the size of squares cut from a box.?

This has been taking me days to do and I really want to do it for test practice. I actually have absolutely no idea how to even start this, so if I can get a hint, advice, or something to start me off,...
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0answers
68 views

Continuation principle

In the following $g$ is a function related with some norms of solution of a certain differential eqution: $g$ be a nonnegative continous (if necessary, it is monotone increasing) function satisfying ...
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1answer
73 views

Convert an inequality into limiting equality.

Given $f(x)/g(x) \lt 1.5/h(x)$ where all three functions are increasing and positive in nature. My question is, if I can deduce $\lim_{x \to \infty} [f(x)/g(x)]=1$ (then how if yes) or not .?
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1answer
60 views

Finding range of transformation of function from range of original

I'm asked to find the range of $y = f(x-2)+4$, if the range of $y=f(x)$ is {$y| -2 \geq y \geq 5, y \in R$}. How do I go about finding this? I have no idea where to even start. I'm doing the course ...
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0answers
384 views

Find a Lipschitz constant

Please help me to find a Lipschitz constant. Let $S_n$ be a group of permutations of the set $\{1, \ldots, n\}$. Let $a=(a_1, \ldots, a_{2M})$ be a real valued vector with $n$ non-zeroes entries, $M \...
-1
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2answers
70 views

What conditions of $x$ such that $e^{x^e} \ge x^{e^x}$? [closed]

What conditions of $x$ such that $e^{x^e} \ge x^{e^x}$ ?
-1
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2answers
116 views

Let $P$ be a polynomial with positive real coefficients. Prove that if $P(1/x) \geq 1/P(x)$ holds for $x = 1$, then it holds for every $x > 0$.

Let $P$ be a polynomial with positive real coefficients. Prove that if $$ P\left( \frac{1}{x} \right) \geq \frac{1}{P(x)} $$ holds for $x = 1$, then it holds for every $x > 0$. What I did: I was ...
-1
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1answer
378 views

Finding a lower bound of a function which is an inequality [closed]

A function $F(n)$ satisfies the recurrence $F(n) \le 7F(3n/2) + 3n$ for all $n \in \mathbb{N}$. Give a lower bound for $F(n)$.
-1
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1answer
73 views

Matries inequality with norms

Let $P$ and $C \neq0$ a $q \times q$ matrices. I want to prove that there exists a positive constants $\alpha$ such under some assumptions under $P$ we have the inequality $${\left\| {P\left( {I - C} ...
-1
votes
5answers
344 views

Midpoint convex functions that are not convex or which are not super-additive but F(0)=0, F injective, and non-negative

When I refer to 'Midpoint Convexity', I mean its form when restricted to the unit interval. I denote this by $(MP)$ . Where, $\text{Dom(F)}=[0,1]$. $$(MP):\forall((x,y)\in [0,1]):\, F\left(\frac{x+y}{...
-2
votes
3answers
78 views

Does $1 + \frac{1}{x} + \frac{1}{x^2}$ have a global minimum for $0 < x \in \mathbb{R}$?

This question is related to this one. My question here is: Does the function $$f(x) = 1 + \frac{1}{x} + \frac{1}{x^2}$$ have a global minimum for $0 < x \in \mathbb{R}$? Thank you!
-2
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2answers
115 views

If $xy+xz+yz=3$ so $\sum\limits_{cyc}\left(x^2y+x^2z+2\sqrt{xyz(x^3+3x)}\right)\geq2xyz\sum\limits_{cyc}(x^2+2)$

Prove that for any set of three positive real $x, y, z$ such that $xy+yz+zx=3$ $x^2(y+z)+y^2(x+z)+z^2(x+y)+2\sqrt {xyz}\left(\sqrt{x^3+3x}+\sqrt{y^3+3x}+\sqrt{z^3+3x}\right)\ge$ $\ge 2xyz(x^2+y^2+z^2+...
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votes
2answers
281 views

Inequality : $\sqrt {x} - 6 - \sqrt{10} -x \geqslant1$ [closed]

I have solved it by squaring both sides and got inequality $x \geqslant 17/2$ but after that, the solution part have concluded on the equation $4x^2 + 289 - 68x \geqslant4(10 - x)$ How this ...