# Questions tagged [functional-inequalities]

For questions about proving and manipulating functional inequalities.

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### Analysis of the functional inequality $f(x-s)f(y+s) \geq f(x)f(y)$

I would like to find references or analysis of the two functional inequalities: $f(x-s)f(y+s)\geq f(x)f(y)$ $f(x-s)+f(y+s) \geq f(x) + f(y)$ where $x,y\in \mathbb{R}$, $y>=x$ and $s>0$. ...
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### Semigroup property between pseudodifferential operators and differential operators

Given a positive integer $n$ and a>0. Let consider the operators $\nabla^n (\cdot)= \sum_{i=1}^d \partial_i^{n}(\cdot)$, and $(1- \Delta)^{\frac a 2}$ defined at the Fourier level as (modulus ...
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### linear transformation of a subset with inequality

Let $F$ be the vector space of all functions $R \to R$. Let $U_2$ be the subset of F formed by all functions bounded by above (a function $f: R \to R$ is bounded from above if there exists $M ∈ R$ ...
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### Prove that $(\frac{\sin x}{x})^3\geq{\cos x}$ with $x\in [0,\, \frac{\pi}{2}]$ [duplicate]

Let $x$ be a real number satisfying $0\leq{x}\leq{\frac{\pi}{2}}$. Prove that $(\frac{\sin x}{x})^3\geq{\cos x}$. I have a problem I need help. The idea of ​​the proof of this problem is to use ...
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### A question about the changes of $f$ when an inequality between $f(x,y)$ and $g(x,y)$ is satisfied

I have two independent, continuous functions that satisfy the following inequality $$f(x,y) \ge g(x,y).$$ Now suppose $g(x,y)$ is set to be fixed, then $x$ and $y$ are constrained. Both functions are ...
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### Appropriate exponent for inequality [closed]

Let consider $1<b<2$, the sets $A=\{ x: |x|\geq2 \} \subset \mathbb{R}^3$, $B=\{x: |x|<2 \} \subset \mathbb{R}^3$ and the expression |x|^{1-b} \chi_{ A} + |x|^{2-b} \chi_{ B ...
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### Bellman-like integral inequality

I have a weakly decreasing, continuous function $w(\cdot)\geq 0$ on $[0,T]$, with $w(0)=1$. I also have a continuous, decreasing function $b(\cdot)$, where $b(t)\geq 0$ for $t\in[0,T]$. I know that \...
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### Popoviciu's inequality

If $f(x): \mathbb R \to \mathbb R$ is a convex function, prove that $$f(x) + f(y) + f(z) + 3 f(\frac{x + y + z}{3}) \geq 2 f(\frac{x + y}{2}) + 2 f(\frac{x + z}{2}) + 2 f(\frac{y + z}{2})$$ I proved ...
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### Proof of inequality $x + x^{-1} - x^r - x^{-r} \geq 0$ for $x>0$ and $0<r<1$.

I am trying to prove the inequality $$x + x^{-1} - x^r - x^{-r} \geq 0$$ for $x>0$ and $0<r<1$. I believe this to be true as it was cited in a proof that I followed, but was not proved there. ...
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### Do injection preserves inequalities?

Suppose $f$ is injective; is it true that $$\Pr[f(x)<y] \qquad \text{is equivalent to} \qquad \Pr[x<f^{-1}(y)]?$$ provided $y\in\mathrm{Ran} f$? What if $f(x)=-x$?
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### Function satisfying $\alpha x(2-x) + (1-\alpha x)g(x) < g(x-\alpha x)$ for all $0\leq x \leq 1$ and $0\leq \alpha <1$.

I am looking for a function $g(x) \in [1, c], c\geq 1$ that satisfies $$\alpha x(2-x) + (1-\alpha x)g(x) < g(x-\alpha x)$$ for all $0\leq x < 1$ and $0\leq \alpha <1$. Across such $g$, I want ...
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### Functional Inequality. Is my approach ok?

Let $f: \mathbb R \to \mathbb R$ be a differentiable function such that $$f(x)^2+f'(x)\le 0.$$ Show that the zero function is the only solution. I started bwoc assuming that there is a point $x$ such ...
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### An inequality considering linear ODE

Let $p(t),q(t)$, and $r(t)$ be continuous functions on an open interval $I$ and let $t_0\in I$. Assume that there exists a positive constant $M$ such that $$|p(t)|+|q(t)|+|r(t)|<M$$ for all $t\in I$...
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### Increasing function $f(x,y,y)=f(x,x,y)$ implies $f(x,y,z)=g(x,z)$?

Got a super super interesting contest problem. $f(x,y,z)$ is a continuous function of three variables; it is required that $x\leq y\leq z$ (this means that $f$ is only defined on the 3-tuples such ...
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### Proving or disproving the functional inequality

I'm trying to prove or disprove the inequality $$\int_{\mathbb{R}}f(x)^{p-2}f'(x)^4dx\leq \frac{3}{p(p-1)}\int_{\mathbb{R}}f(x)^pf''(x)^2dx$$ for any real-valued function $f$ (provided the RHS is ...
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### Solve the functional inequality $f(x+1)>(f(x))^2-f(x)+1$
Suppose $f(x):[0,\infty)\longrightarrow [0,\infty)$ satisfies $$f(x+1)>(f(x))^2-f(x)+1$$ for sufficiently large values of $x$. Can we solve this functional inequality or get some good information ...