Mathematics

Questions tagged [integral-inequality]

For questions inequalities which involves integrals, like Cauchy-Bunyakovsky-Schwarz or Hölder's inequality or Minkowski's inequality .

Integral inequalities are mathematical inequalities where integrals appear as part of the expressions being compared. These inequalities are fundamental tools in analysis, with applications ranging from proving convergence and estimating bounds to functional analysis, probability theory, and partial differential equations.

Some of the most common integral inequalities include:

  • Cauchy–Bunyakovsky–Schwarz inequality:
    $ \left( \int_a^b f(x)g(x)\,dx \right)^2 \leq \left( \int_a^b f(x)^2\,dx \right) \left( \int_a^b g(x)^2\,dx \right). $

  • Hölder’s inequality:
    $ \int |fg| \leq \left(\int |f|^p\right)^{1/p} \left(\int |g|^q\right)^{1/q}, \qquad \frac{1}{p} + \frac{1}{q} = 1. $

  • Minkowski’s inequality (triangle inequality for $L^p$-norms):
    $ \left( \int |f+g|^p \right)^{1/p} \leq \left( \int |f|^p \right)^{1/p} + \left( \int |g|^p \right)^{1/p}. $

  • Jensen’s inequality (for convex functions and probability measures).

Usage guidance:

  • Use this tag when the main focus of the problem is an inequality involving integrals.
  • If the inequality does not involve integration, prefer the general [inequality] tag.

Examples of typical questions:

  • Proving a classical inequality (e.g., Hölder’s or Minkowski’s).
  • Establishing bounds on integrals of functions.
  • Applying integral inequalities to series, norms, or expectations in probability
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Is this continuous analogue to the AM–GM inequality true?

First let us remind ourselves of the statement of the AM–GM inequality: Theorem: ( AM–GM Inequality ) For any sequence $(x_n)$ of $N\geqslant 1$ non-negative real numbers, we have $$\frac1N\sum_k x_k \geqslant \left(\prod_k x_k\right)^{\frac1N}$$ It…
user1892304
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Integral Inequality Absolute Value: $\left| \int_{a}^{b} f(x) g(x) \ dx \right| \leq \int_{a}^{b} |f(x)|\cdot |g(x)| \ dx$

Suppose we are given the following: $$\left| \int_{a}^{b} f(x) g(x) \ dx \right| \leq \int_{a}^{b} |f(x)|\cdot |g(x)| \ dx$$ How would we prove this? Does this follow from Cauchy Schwarz? Intuitively this is how I see it: In the LHS we could have a…
fourierguy
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Prove the following integral inequality: $\int_{0}^{1}f(g(x))dx\le\int_{0}^{1}f(x)dx+\int_{0}^{1}g(x)dx$

Suppose $f(x)$ and $g(x)$ are continuous function from $[0,1]\rightarrow [0,1]$, and $f$ is monotone increasing, then how to prove the following inequality: $$\int_{0}^{1}f(g(x))dx\le\int_{0}^{1}f(x)dx+\int_{0}^{1}g(x)dx$$
Larry Eppes
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On the inequality $ \int_{-\infty}^{+\infty}\frac{(p'(x))^2}{(p'(x))^2+(p(x))^2}\,dx \le n^{3/2}\pi.$

$ p(x)\in\mathbb{R[X]} $ is a polynomial of degree $n$ with no real roots. Show that: $$\int\limits_{-\infty}^{+\infty}\dfrac{(p'(x))^2}{(p'(x))^2+(p(x))^2}\,dx \leq n^{3/2}\pi.$$ It's easy to see that the degree of $ p$ has to be even. For $n=2$…
jack
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Geometric interpretation of Hölder's inequality

Is there a geometric intuition for Hölder's inequality? I am referring to $||fg||_1 \le ||f||_p ||g||_q $ , when $\frac{1}{p}+\frac{1}{q}=1$ . For $p=q=2$ this is just the Cauchy-Schwarz inequality, for which I have geometric intution: The…
Asaf Shachar
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Prove that $\left|30240\int_{0}^{1}x(1-x)f(x)f'(x)dx\right|\le1$.

Let $f\in C^{3}[0,1]$ such that $f(0)=f'(0)=f(1)=0$ and $\big|f''' (x)\big|\le 1$ .Prove that $$\left|30240\int_{0}^{1}x(1-x)f(x)f'(x)dx\right|\le1 .$$ I couldn't make much progress on this problem. I thought that maybe I should try using polynomial…
JustAnAmateur
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Jensen's inequality for integrals

What nice ways do you know in order to prove Jensen's inequality for integrals? I'm looking for some various approaching ways. Supposing that $\varphi$ is a convex function on the real line and $f$ is an integrable real-valued function we have that:…
user 1591719
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Proof of Wirtinger inequality

Quoting from Ana Cannas da Silva's book on Symplectic Geometry: "As an exercise in Fourier series, show the Wirtinger inequality: for $f\in C^1([a,b])$, with $f(a)=f(b)=0$ we have $$ \int_a^b\Big|\frac{\mathrm{d}f}{\mathrm{d}t}\Big|^2\mathrm{d}t…
Brightsun
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Prove that:$f(f(x)) = x^2 \implies \int_{0}^{1}{(f(x))^2dx} \geq \frac{3}{13}$

Let $f: [0,\infty) \to [0,\infty)$ be a continuous function such that $f(f(x)) = x^2, \forall x \in [0,\infty)$. Prove that $\displaystyle{\int_{0}^{1}{(f(x))^2dx} \geq \frac{3}{13}}$. All I know about this function is that $f$ is bijective, it is…
C_M
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A tricky integral inequality

A friend has submitted this problem to me: Let $0<a<b<1$ and $f:[0,1]\to \mathbb R$ be a differentiable function such that $$\displaystyle \frac{\int_0^a f(x) dx}{a(1-a)}+\frac{\int_b^1 f(x) dx}{b(1-b)}=0$$ Prove that $\displaystyle \left| \int_0^1…
Gabriel Romon
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Reverse Cauchy Schwarz for integrals

Let $f,g$ be two continuous positive functions over $[a,b]$ Let $m_1$ and $M_1$ be the minimum and maximum of $f$ Let $m_2$ and $M_2$ be the minimum and maximum of $g$ Prove that $$\sqrt{\int_a^bf^2 \int_a^b g^2}\leq…
Gabriel Romon
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Do inequalities that hold for infinite sums hold for integrals too?

Let $\mathbb{R}_{\geq0}$ denote the set of non-negative reals and $+\infty$, and $\mathbb{Z}^+$ denote the set of positive integers. I will also let $\lambda$ denote the Lebesgue measure on $\mathbb{R}$ . Let there be a function…
Amr
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How prove this $\int_{a}^{b}f^2(x)dx\le (b-a)^2\int_{a}^{b}[f'(x)]^2dx$

let $f\in C^{(1)}[a,b]$,and such that $f(a)=f(b)=0$, show that $$\int_{a}^{b}f^2(x)dx\le (b-a)^2\int_{a}^{b}[f'(x)]^2dx\cdots\cdots (1)$$ My try: use Cauchy-Schwarz inequality we have $$\int_{a}^{b}[f'(x)]^2dx\int_{a}^{b}x^2dx\ge…
math110
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Holder's inequality for infinite products

In analysis, Holder's inequality says that if we have a sequence $p_1, p_2, \ldots, p_n$ of real numbers in $[1,\infty]$ such that $\sum_{i=1}^n \frac{1}{p_i} = \frac{1}{r}$, and a sequence of measurable functions $f_1, f_2, \ldots, f_n$, then…
JHF
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An integral inequality (one variable)

Anyone has an idea to prove the following inequality? Let $g:\left(0,1\right)\rightarrow\mathbb{R}$ be twice differentiable and $r\in\left(0,1\right)$ such that $$…
Binjiu
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