# Questions tagged [integral-inequality]

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

1,085 questions
Filter by
Sorted by
Tagged with
118 views

34 views

89 views

### How to estimate $\int_0^{L/2} f(x)dx$ from $f(x)\ge 2x-\frac{L}{D} \int_0^x f(t)dt$

$D>0, L>0$ are constant. $f:[0, +\infty) \rightarrow \mathbb R$ is non-negative continuous function. $f(0)=0$ and $D=\int_0^{+\infty} f(x) dx$. In fact, when $x>0$ is large enough (than $L$),...
31 views

### Prove that $\left(\int_0^1 f(x)dx\right)^2\leq\frac{1}{12}\int_0^1\left(f'(x)\right)^2dx$ [duplicate]

The following problem is from Andreescu's "Problems in Real Analysis: Advanced Calculus on the real axis". More specifically it's problem $10.6.11$: Let $f:[0,1]\rightarrow\mathbb{R}$ be a ...
139 views

61 views

537 views

683 views

### Upper bound on an integral

Let $f : [0,1] \to \mathbb{R}$ be a function satisfying: 1.) $|f(x)| \leqslant a$ for some $a < 1$, and 2.) $\int_0^1 f(x) dx = 0$. I would like to know whether the following inequality ...
345 views

1 vote
105 views

### If $g:[a,b] \to \mathbb{R}^m$ is continuous and $x_0 \in [a,b]$, prove $|\int_a^b g(x)dx| \geq \int^a_b|g(x)|dx-2\int^a_b|g(x)-g(x_0)|dx$.

This is an exercise from Leoni’s book on Sobolev spaces. It seems elementary but the proof is eluding me despite my repeated attempts to use the reverse triangle inequality and the mean value theorem. ...
61 views

### Proving $\left\vert\int_n^{n+1}s\int_n^xt^{-s-1}dt\,dx\right\vert<\vert sn^{-s-1}\vert$, for $n \in \mathbb{N}$, $x \in [n;n+1]$, $s\in \mathbb{C}$

Let's get out of silence? I try to prove the Riemann hypothesis because I have the humble intuition that it is true, whatever you may think of this step, with my little skills. I have time, so... I ...
52 views

### 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 \...
1 vote
72 views

### Integral inequality $\left| \int\limits_{-1}^{1} x f(x) dx \right| \le \left| \int\limits_{-1}^{1} f(x) dx \right|$

I need to find the widest class I can for functions that obey the following inequality: $$\left| \int\limits_{-1}^{1} x f(x) dx \right| \le \left| \int\limits_{-1}^{1} f(x) dx \right|$$ It is clear ...
1 vote
78 views

97 views

Let $\gamma : S^1 \rightarrow M$ be a smooth map from a circle of length 1 to a closed manifold $M$ with nonpositive curvature. Could we find a constant $C > 0$ depending only on $M$ such that $$\... 1 vote 0 answers 112 views ### Upper Bound Product of Integrals I have an explicit pdf p(x) that I want to show is log-concave, i.e. \log p(x) is a concave function. Under suitable assumptions, this is equivalent to showing that p(x) satisfies the inequality ... 2 votes 1 answer 64 views ### Integral inequality I_2(z) = \mathcal{O}\left(\frac{1}{(1+z)^2}\right) I want to prove the inequality of an integral like this:$$I_2(z) = \int_1^{+\infty} \frac{\rho(t)}{(z+t)^2} d t = \mathcal{O}\left(\frac{1}{(1+z)^2}\right)$$, when z \to +\infty Below is my ... 0 votes 1 answer 78 views ### A variant of Cauchy-Schwarz Inequality for integral I have encountered the above inequality in a proof, where u(\tau) \in \mathbb{R}^{n\times m}, and w \in \mathbb{R}^n is any unit vector T_0 and t is are positive real numbers The proof ... 0 votes 2 answers 29 views ### How to bound m(A)^{-1}\int_A|f(x) - c|^tdx above by \left(m(A)^{-1}\int_A|f(x) - c|dx\right)^t for 0 < t < 1 and A a compact set Let m denote the Lebesgue measure on \mathbb{R}^n, f be a locally integrable function in \mathbb{R}^n, let 0 < t < 1, A\subset\mathbb{R}^n be a compact subset and c some finite ... 0 votes 0 answers 31 views ### Relationship between bounding volume integral and surface integral Assume we have a (d-1)-dimensional surface S and a sequence of precompact, decreasing open sets R_{n+1} \subseteq R_{n} \subseteq \mathbb{R}^d containing S such that \cap_{n\in\mathbb{N}} R_n ... 3 votes 0 answers 138 views ### Minkowski's integral inequality for other norms In my measure theory course we studied norms L_p and no other norms. For proofs we used exclusively the trick known as Hoelder's inequality which works only on L_p norms. I disliked it very much ... 3 votes 0 answers 46 views ### Inequality with integral and distribution function I encountered an inequality with the following variables and functions. X is a random variable drawn from (-\infty, \infty) with cdf F, pdf f, and mean \mu=\mathbb{E}[X]. For any x and \... 0 votes 1 answer 80 views ### Suppose \int_{0}^{a}f(x)dx = 1. Is it true that for every b>0 we have \int_{0}^{a}\frac{e^{bx}}{b}\left(f(x)\right)^2dx \ge 1 \; ?  Suppose a>0 is such that$$ \int_{0}^{a}f(x)dx = 1. $$Is it true that for every b>0 we have$$ \int_{0}^{a}\frac{e^{bx}}{b}\left(f(x)\right)^2dx \ge 1 \; ? 1 vote 0 answers 111 views ### Reverse Poincaré inequality and Fourier Transform? My advisor gave me the other day a draft of a paper he's been working on to study. The following is only a part of a proof which I am trying to figure out at the moment. Now since, I'm afraid of ... 0 votes 0 answers 36 views ### an integral inequality involving determinant I have a matrix of smooth functions (f_{ij}) on a compact manifold M, and the equality \int_M det(f_{ij})\ d\omega = 1; is it possible to get some lower bound on det(G_{ij}), where G_{ij} = \... 0 votes 1 answer 39 views ### Inequality about integral of probability density function in minimax theory Recently I have been reading some material about minimax theory and there is an inequality that I don't understand how to derive it. That is \begin{align*} \int[p_0^n(x)\land p_1^n(x)]dx \geq\frac{1}{... 3 votes 0 answers 64 views ### How to prove this integral inequality by variational method？ Suppose f(x) is fourth differentiable and satisfies f(0)=f(1)=f'(0)=f'(1)=0.Prove the inequality\frac{1}{a^4}\int^{1}_{0}[f''(x)]^2\text dx\geq \int^{1}_{0}[f(x)]^2\text dxWhere $a$ is the ...
1 vote
In Wikipedia https://en.wikipedia.org/wiki/Gr%C3%B6nwall%27s_inequality#Integral_form_for_continuous_functions we can find the following statement: Let $I$ denote an interval of the real line of the ...
I am reading the following proof of integral inequality lemma: Let $b(t)$ and $f(t)$ be continuous functions for $t \geq \alpha$, and $v(t)$ is a differentiable function for $t \geq \alpha$. Suppose ...