# Tagged Questions

For questions inequalities which involves integrals, like Cauchy-Bunyakovsky-Schwarz or Hölder's inequality. To be used with (inequality) tag.

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### Prove or disprove this inequality

Let $p, q, a$, and $b$ be natural numbers such that $p<q$, $1<b<a$ and $b\nmid a$. Is is true that $(bp+aq)^3> (a^3+b^3)q^3$? This is what I tried: expanding the left-hand side, we ...
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### Determining Bounds to calculate mass

Let $E$ be the solid region defined by the inequalities $x \ge 0$, $0\le z \le \sqrt(x^2 + y^2)$, $x^2 + y^2 + z^2 \le 4$ Suppose that $E$ has mass density $\mu(x,y,z) = xz$. Calculate the ...
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### Inequality: product of integrals

Context: Proving integral inequalities about posterior distributions following different sequences of binary signals. The proofs come down to the following inequalities. Let $\psi(x)$ be a concave ...
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### 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 ...
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### Rewrite order of $\int_0^1\int_0^{\sqrt{y}} \int_y^1 \, dz \, dx \, dy$ to $dx\,dy\,dz$ and $dy\,dz\,dx$

I need to change the order of $$\int_0^1\int_0^{\sqrt{y}}\int_y^1\,dz\,dx\,dy$$ to $dx\,dy\,dz$ and $dy\,dz\,dx.$ I can extract the inequalities to get $1≤z≤y$, $0≤x≤\sqrt y$, $0≤y≤1$, but I get ...
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### Inequality releating squared absolute value of an integral to the integral of the squared absolute values of the integrand

Is this inequality $\left| \int_{0}^{x} f(t) \ dt \right|^2 \leq \int_{0}^{x} |f(t)|^2 \ dx$ true for $x\in [0,1]$. In case it is how to prove it? If there is no square in both sides it is easy since ...
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Let $\psi \in C_0^{\infty}(\mathbb{R}^3)$. How to prove (or where I can find this proof) that $$\int_{\mathbb{R}^3}\frac{1}{4r^2}|\psi(x)|^2d^3x\le \int_{\mathbb{R}^3}|\nabla\psi(x)|^2d^3x$$ ? ...
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### Proving a Definite Integral Inequality without Geometrical Intuition

I solved an integral inequality problem using geometrical methods. However, I just cannot satisfy with them and want a without-geometrical-intuition proof, and I couldn't find one. Proof the ...
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### Integral and differential inequality

I have integral and differential inequality $y'(t)<Ch^{k+1}+\int_0^ty(s)ds+y(t)$ where $C,h$ are constants and $y$ is positive function with y(0)=0 My goal is to prove $y(t_F)<Ch^{k+1}$ ...
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### A Young's inequality used to bound curvature terms

I've been having a look at the Gage and Hamilton's The Heat Equation Shrinking Convex Plane Curves (here). In particular I've been working in the Lemma 4.4.2 and some further results where they find ...
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### integral inequality for $f(x)$ and $f(\sqrt{x})$

Show that if $f(x)\in [0;1]$, $f\in C$ and $\int\limits_{1}^{+\infty}f(t)dt=A$ then $\int\limits_{1}^{+\infty}tf(t)dt>\frac{A^2}{2}$ I only have noticed two small things: If $A=1$ inequality is ...
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### Kind of Gronwall Inequality

Does somebody knows if it is possible to obtain an inequality (like for Gronwall inequality) on $f$ if $f$ verify $$f(t) \leq A+\int_0^{2t} g(s)f(s) ds$$. Where $f$ and $g$ are as smooth as ...
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### For what value of $x$: $n^ {(x+1)} + n^ {2x} < n^2$ ? Where, $0\leq x <1$ and $n$ is constant integer value & $n>1$.

How to find the optimal value of $x$ and what is the relation between $x$ and $n$ i.e. How to get dependency between $x$ and $n$? As per my understanding, solution should be in term of $n$ like like ...
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### Integral inequality involving $f(x),\, x\,f(x),\, f(x)^2$

Let $f\colon[-1,1]\to\mathbb{R}$ be a continuous function. Prove that $$2\int_{-1}^{1}f(x)^2\: dx - \left(\int_{-1}^{1}f(x)\: dx\right)^2 \ge 3\,\left(\,\int\limits_{-1}^{1}x\,f(x)\: dx\right)^2$$ ...
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### Name or reference about a inequality with integrals?

I have wrote down some class notes and I think I copied something wrong. It is an integral inequality; $$\iiint_{B^n}|\nabla\psi|^2\frac{1}{|x|^{n-2}}dV\leq C\iint_{\partial B^n}|\psi|^2dA$$ where $C$...
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### Is the Riemann integral of a strictly smaller function strictly smaller?

We all know that if $f\leq{}g$ in $[a,b]$ then $$\int_a^bf\,dx\leq\int_a^bg\,dx$$ now, imagine that we have $f<g$, is it true that $$\int_a^bf\,dx<\int_a^bg\,dx$$
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### Explain inequality of integrals by taylor expansion

I try to understand why the following inequality holds. $$\left|\int_{|y|<1} e^{iuy}−1−iuy\ \, dy \right| \le \frac{1}{2} \cdot \int_{|y|<1} |uy|^2\ \, dy$$ Due to a hint I'm pretty sure, ...
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### Relation between Karamata's and Hardy-Littlewood's inequalities

In the field of (elementary) classical inequalities one of the most famous tools is the majorization inequality due to Karamata [1] (also known as Hardy-Littlewood-Polya). In its integral version, it ...
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### Prove that $e^x|\int_x^{x+1}\sin(e^t)dt|\le 2$ [duplicate]

Prove that $e^x|\int_x^{x+1}\sin(e^t)dt|\le 2$. Use mean value theorem $$\int_x^{x+1}\sin(e^t)dt=\sin(e^\xi)$$ And we have $$|\sin(e^\xi)|\le\frac{2}{e^x}$$ where $\xi\in(x,x+1)$ I stuck here. ...
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### For what values of $0 < p,q < \infty$ is the following inequality of integrals valid?

Let $m$ be the Lebesgue measure over $\mathbb{R}$ and let $f$ and $g$ be two nonnegative measurable functions defined on $[0,1]$ such that $f(x)g(x)\geq 1 \quad \forall x \in [0,1]$. It is not ...
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### Inequality verification of the ratio of two integrals involving Bessel functions

Given the following integral: $\sigma(k,\theta)=2k^2cos^2\theta\int_0^\infty J_0(2k\tau |sin\theta|) exp(-2s^2k^2\tau^{2H}cos^2\theta)) \tau d\tau$ With the following constraints $0.5<H<1$...
### How to prove that: if $q= b+d$, then $p = a+c$?
Let $a,b,c,d,p$, and $q$ be natural numbers such that $ad-bc = 1$ and $\frac{a}{b} > \frac{p}{q} > \frac{c}{d}$. How to prove that: if $q= b+d$, then $p = a+c$? Is there a simple way?