Questions tagged [integral-inequality]

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

772 questions
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Maximum of $\int_0^1(x^p|f(x)|^q-x^q|f(x)|^p)dx$

Let $p>q>0$ and $C=\{f:[0,1] \to \mathbb{R} \mid f \text{ is continuous} \}$. Determine $$\max_{f \in C}\int_0^1(x^p|f(x)|^q-x^q|f(x)|^p)dx$$ and the functions for which this maximum occurs. ...
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Prove that $\int_0^1 (f'(x))^2dx \geq \frac{30}{31}$ when $f(f(x))=x^2$

Let $f:[0, \infty) \to [0,\infty)$ be a differentiable function with $f'$ continuous. If $f(f(x))=x^2$, prove that $$\int_0^1 (f'(x))^2dx \geq \frac{30}{31}$$ without explicitly finding $f.$ Since we ...
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$f,g \in [0,1] \times [0,1]$, $\int f - g \mathrm{d}x = 0$ and are monotonically increasing, then $\int |f-g| \mathrm{d}x \le \frac{1}{2}$

$f,g$ are monotonically increasing in $[0,1]$ and $0\le f , g \le 1$. $\int_0^1 f - g \mathrm{d}x = 0$. Prove that $$\int_0^1 |f - g|\mathrm{d}x \le \frac{1}{2}$$ In my previous question, $g(x) = x$....
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Proof-Verification: $\int_a^b \left(\frac{b-x}{b-a}\right)^{2018}f(x)\,{\rm d}x \leq \frac{1}{2019}\int_a^b f(x)\,{\rm d}x$

Problem Let $f(x)$ be continuous and increasing over $[a,b]$. Prove $$\displaystyle \int_a^b \left(\frac{b-x}{b-a}\right)^{2018}f(x){\rm d}x \leq \frac{1}{2019}\int_a^b f(x){\rm d}x.$$ Proof By ...
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Is it always true that integral of nonnegative function is non negative [closed]

If $f(x)\geq 0$, is it true that $\int f(x)dx \geq 0$?
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$\int_a^bf^2(x)\,dx\le \frac{2}{3}\int_a^bf(x)\,dx$ for a convex differentiable function

If $f:[a,b] \to \mathbb{R}, f(a)=0,f(b)=1$ is a convex increasing differentiable function on the interval $[a,b]$ . Prove that $$\int_a^bf^2(x)\,dx\le \frac{2}{3}\int_a^bf(x)\,dx$$ Since f is ...
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Can we include inequalities while solving under determined simultaneous linear equation in reduced Echelon form?

Is there a way to include the inequalities of variables in calculating family of equations while solving under determined simultaneous linear equations? For Eg: Lets say x + y = 4, y + z = 4. But ...
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How to find the sharp constant between norms?

How to prove that $\|u\|_\infty\leq C_p\|u'\|_{L^p},\ \forall\ u\in W^{1,p},\ u(0)=u(T)$ with $\int_0^T u=0$ and $C_p=\frac{1}{2}\left[\frac{T(p-1)}{2p-1}\right]^{\frac{p-1}{p}}$? It is easy to show ...
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Application of Gronwall Inequality to existence of solutions

Consider the $N$-dimensional autonomous system of ODEs $$\dot{x}= f(x),$$ where $f(x)$ is defined for any $x \in \mathbb{R}^N$, and satisfies $||f(x)|| \leq \alpha||x||$, where $\alpha$ is a ...
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Integration inequality, why can we pull e^t out of the integral and leave its e?

In the following, at line 3 $e^t \sin(t)$ is pulled out of the $|\cdot|$ and left as a constant. How does one justify this step?
In the proof of existence/uniqueness of SDE the following inequality is used: $$E\left[ \left( \int_0^t a(s,\omega) ds \right)^2 \right] \leq t E\left[ \int_0^t a(s,\omega)^2 ds \right]$$ and I ...
Compute the limit $\lim_{n\to\infty} I_n(a)$ where $I_n(a) :=\int_0^a \frac{x^n}{x^n+1}\,\mathrm{d}x, n\in N$.
For $a>0$ we define $$\space I_n(a)=\int_0^a\frac{x^n}{x^n+1}\,\mathrm{d}x , n\in N.$$ Prove that $0\le I_n(1) \le \frac{1}{n+1}$ Compute $\lim_{n\to\infty} I_n(a)$ My attempt: I regard \$I_n(1)=...