# Questions tagged [integral-inequality]

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

764 questions
99 views

37 views

### 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. ...
75 views

### 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 ...
65 views

### 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 ...
46 views

### $f,g \in [0,1] \times [0,1]$, $\int f - g \mathrm{d}x = 0$ 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, $g(x) = x$....
37 views

22 views

### prove inequality with an integral over region with unit length

I am trying to show that $\log{m}\le{\int_{m}^{m+1}{\log{t}}}dt$, with $m\ge{1}$. I tried simplifying the problem to $0\le{\int_{m}^{m+1}{\log{\big(\frac{t}{m}\big)}}}dt$, but can't seem to get any ...
67 views

15 views

### Linear version of Gronwall's inequality, proof

I am reading a proof of the following theorem: Assume $\phi$ is a continuous function in $[0,T]$ that satisfies $$\phi(t) = \alpha + \int_0^t (\beta \phi(s) + \gamma)ds, \hspace{0.5mm} t\in [0,T],$$ ...
30 views

### Help bounding a “norm”

In Weak Convergence and Stochastic Processes, the authors introduce the following notation: $$\|\xi\|_{2,1} = \int_0^\infty \sqrt{P(\xi > x)}\,\mathrm dx$$ They then admit that this is technically ...
2k views

### Lyapunov's inequality in Probability

I have a question about the proof of the inequality. The well known result stats Let $Z$ be a RV and let $0<s<t$. Then $$E(|Z|^s)^{1/s} \leq E(|Z|^t)^{1/t}$$ The proof follows almost ...
133 views

98 views

154 views

### Show that $\int_{0}^{\pi/6} {\cos (x^2)}\mathrm{d}x\ge\frac12$.

Prove that $\displaystyle\int_{0}^{\frac\pi 6} {\cos ({x^2)}\mathrm{d}x\ge\dfrac12}$. I know this is a Fresnel integral but without going into advanced calculus is there a way to show that this is ...
38 views

### Prove that if $f \in L^p$ and $g \in L^q$ where $p$ and $q$ are conjugate exponents , then $\lim_{\vert x \vert \to \infty}(f*g)(x)=0$

The convolution of $f$ and $g$ on $R^d$ equipped with the lebsgue measure is defined by $$(f*g)(x)=\int_{R_d} f(x-y)g(y) \, dy$$ Prove that if $f \in L^p$ and $g \in L^q$ where $p$ and $q$ are ...
104 views

### How to prove that $\int\limits_0^{\pi} e^{\sin^2(x)}dx > {3\over2}\pi$? [closed]

How to prove that $\int\limits_0^{\pi} e^{\sin^2(x)}\ dx > {3 \over 2}\pi$?
I tried to prove that $$\int_{a}^{b}\frac{|\cos (x)|}{x}dx\leq \frac{2}{\pi}\log\left(\frac{b}{a}\right)+O(1).$$ Of course $O(1)$ as a function of $b$, i.e. a bounded function of $b$. $a$ is ...
Let $0<r<1$ and $t\geq 0$ real numbers. Is it true that $$\int_t^{t+r} \sin(x)\, dx \leq \int_{\frac{\pi}{2}-\frac{r}{2}}^{\frac{\pi}{2}+\frac{r}{2}}\sin(x)\, dx \,?$$ I suspect that yes, ...