Questions tagged [holder-inequality]

Proving or manipulations with inequalities by using Hölder's inequality.

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Are these two Hölder-type inequalities valid?

Hereinafter, we use $p$-norm ($p\geqslant1$) in $n$-dimensional Euclidean space, which is given as $\|x\|_p=(\sum_{i=1}^n|x_i|^p)^{1/p}$. As we know that, the following inequality holds for all $x\in\...
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Show the inequality $\sum_{\text{cyc}} \sqrt{\frac{a^3}{b^3 + (c+a)^3}} \ge 1$

For all positive reals $a, b, c$, we wish to prove the inequality $$\sum_{\text{cyc}} \sqrt{\frac{a^3}{b^3 + (c+a)^3}} \ge 1$$ My approach was Hölder: $$\left(\sum_{\text{cyc}} \sqrt{\frac{a^3}{b^3 + (...
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Prove an integral inequality with squared integrals

Given $f, g$ integratable prove that $$\left(\int_0^1 f(t) \ \mathrm{d}t\right)^2 + \left(\int_0^1 g(t) \ \mathrm{d}t\right)^2 \leq \left(\int_0^1 \sqrt{f^2(t) + g^2(t)} \ \mathrm{d}t\right)^2$$ I ...
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Bound on convolution: $ | (h * f^2) (x)| \leq \| f\|^2_2 g(h)$

I am trying to find bounds for the following quantity. Take two functions $f,h \in L^{1} \cap L^2$ but $\|h \|_{\infty} = \infty$. Is there a way to obtain a bound of the following type: $$ | (h * f^2)...
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Integral estimate with Young's inequality

Consider two functions $g,h$ with $g\in L^m(\Omega)$ and $h\in L^{l+\epsilon}(\Omega)$, where $\Omega\subset\mathbb{R}^l$ ($l\geq 2$) is some bounded domain, $\epsilon>0$ is small (for simplicity ...
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Equation $x^4+ax^3+bx^2+ax+1=0$ has at least 1 real root. $a,b \in \mathbb{R}$ What's the minimum value of $(a^2+b^2)$? [duplicate]

Equation $x^4+ax^3+bx^2+ax+1=0$ has at least 1 real root. $a,b \in \mathbb{R}$ What's the minimum value of $(a^2+b^2)$? Ok so here I tried to divide by $x^2$ and get this : $$x^2+\frac{1}{x^2}+a\left(...
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Does this inequality for sums hold for integrals too?

Let $f,g:I \to \mathbb R$ be positive functions on a finite set, then $$ \left(\sum_i f(i)\right)\left(\sum_i g(i)\right) \geq \sum_i f(i)g(i) $$ The proof is rather trivial: by multiplying out the ...
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Understanding the Theorem in Sec . 7.8. of Titchmarsh's Book

The following is a part of the theorem in Section 7.8. in Titchmarsh's book The Theory of the Riemann Zeta-Function: My questions: 1- I need to study Parseval's formula but lets say that the formula $...
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Comment on the uniqueness of the solution of ODE

Consider the first-order equation given by $y' = f(x, y)$ in $I := [x_0, b]$ subject to the initial condition $y(x_0) = y_0$, assuming $f$ is continuous everywhere it is defined and there exists $M &...
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Ratio involving $l_p$ norms

Let $w$ be an $n$-dimensional real vector with non-negative entries, and let $t>1$. I want to bound $$\frac{\sum_{i=1}^n w_i \times \sum_{i=1}^n w_i^{2t-1}}{(\sum_{i=1}^n w_i^t)^2}$$ In the ...
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Minimal order of $\bigg|\sum\limits_{k=1}^na^3_k\bigg|$ given $\bigg|\sum\limits_{k=1}^na_k\bigg|\geq\sqrt{n}$

Does there exist a sequence of real numbers $\{a_k\}_{k=1}^\infty$ such that for every $n\in\mathbb{N}$ we have $$\bigg|\sum\limits_{k=1}^na_k\bigg|\geq\sqrt{n}$$ but $$\bigg|\sum\limits_{k=1}^na^3_k\...
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Taking the limit in Holder's inequality

I have a standard normal random variable $X\sim\mathcal{N}(0,1)$ and an event $E$ with $\mathbb{P}(E)=p$, and this event is about $X$ and some other variables. I am interested in upper-bounding the ...
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Lyapunov inequality and $L^p$ interpolation

I am trying to prove the this called "Lyapunov's inequality": $$\big\||f_0|^{p_0(1-\theta)/p}|f_1|^{p_1\theta/p}\big\|^p_p\leq\|f_0\|^{(1-\theta)p_0}_{p_0}\|f_1\|^{p_1\theta}_{p_1}$$ which ...
Physics user's user avatar
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For which $p\in [1,+\infty]$ is the following operator continuous?

For each $n\in \Bbb N$, $n\ge1$ and for each $x\in \Bbb R$, consider the following operator: $T_n: L^2(\Bbb R) \to L^p(\Bbb R) $ defined as $$T_nf(x)=n^{3/4} \int_x^{x+1/n} f(t)\mathrm dt $$ The ...
Sine of the Time's user avatar
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Showing $\frac{1}{5}<(f(k,l,m,n))^3\leq\frac{64}{11}$, where $f(k,l,m,n)=\sum_{cyc}\frac{k^2}{\sqrt[3]{5k^6+6l^3m^3}}$ for positive $k$, $l$, $m$, $n$

For $k,l,m,n \in \mathbb{R}^+$ denote by $f(k,l,m,n)$ the following function: $$f(k,l,m,n) = \frac{k^2}{\sqrt[3]{5k^6+6l^3m^3}}+\frac{l^2}{\sqrt[3]{5l^6+6m^3n^3}}+\frac{m^2}{\sqrt[3]{5m^6+6n^3k^3}}+\...
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If $u \notin L^p[0,1]$, can we find some $w \in L^{\frac{p}{p-1}}[0,1]$ such that $\int_0^1 u \cdot w = \infty$?

The question is as in the title. For some fixed $p \in (1,\infty)$, let $p' \in (1,\infty)$ be such that $\frac{1}{p}+\frac{1}{p'}=1$. Then, for any $u \in L^1[0,1] - L^p[0,1]$, I wonder if it is ...
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$\frac{a}{\sqrt{2a^2+3bc}}+\frac{b}{\sqrt{2b^2+3ca}}+\frac{c}{\sqrt{2c^2+3ab}} \le \sqrt{ab+bc+ca}$

Given non-negative real numbers $a,b,c$ satisfying $a+b+c=3$.Prove that $$\frac{a}{\sqrt{2a^2+3bc}}+\frac{b}{\sqrt{2b^2+3ca}}+\frac{c}{\sqrt{2c^2+3ab}} \le \sqrt{ab+bc+ca}.$$ By C-S we need proof $$4(...
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3 votes
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How to use Lebesgue Number for Complex Analysis Holder Inequality

Problem: Let $f$ be analytic in an open set $U\subset \mathbb{C}$ and let $K\subset U$ be compact. Show that there exists a constant $C$ depending on $U$ and $K$ such that $$|f(z)| \leq C \left( \...
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2 votes
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If $a^2+b^2+c^2=9$ prove $\frac{1}{\sqrt[3]{a^3+b^3}}+\frac{1}{\sqrt[3]{c^3+b^3}}+\frac{1}{\sqrt[3]{a^3+c^3}}\ge \sqrt[3]{\frac{a+b+c}{2}}. $

Given positive real numbers $a,b,c$ satisfying $a^2+b^2+c^2=9.$ Prove that$$\color{black}{\frac{1}{\sqrt[3]{a^3+b^3}}+\frac{1}{\sqrt[3]{c^3+b^3}}+\frac{1}{\sqrt[3]{a^3+c^3}}\ge \sqrt[3]{\frac{a+b+c}{...
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Hoelder's inequality for measurable functions alone?

Are their examples where hölders inequality can't be applied to just measurable functions i.e. if $f$ and $g$ are measurable and $fg$ integrable, but $f$ may be not in $L^p$ or $g$ may be not in $L^q$ ...
MackeyTopology's user avatar
4 votes
2 answers
169 views

Conjecture on the representation of $\log$-convex functions

let $U$ be a convex open set, a function $f \; : \; U \to \mathbb{R}^{+}$ is said to be $\log$-convex when $$f\left(\frac{x}{p}+\frac{y}{q}\right) \leq f(x)^{1/p}f(y)^{1/q}$$ for any $x,y \in U$ and $...
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Inequality $\frac{1}{\sqrt{ab+2}}+\frac{1}{\sqrt{bc+2}}+\frac{1}{\sqrt{ca+2}}\ge \frac{2\sqrt{3}}{\sqrt{a+b+c+abc}}.$

Let $a,b,c\ge 0: ab+bc+ca=3.$ Prove that $$\frac{1}{\sqrt{ab+2}}+\frac{1}{\sqrt{bc+2}}+\frac{1}{\sqrt{ca+2}}\ge \frac{2\sqrt{3}}{\sqrt{a+b+c+abc}}.$$ Equality occurs when $a=b=c=1.$ Also, there's an ...
Dragon boy's user avatar
2 votes
1 answer
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Inequality $\frac{1}{\sqrt{4a^2+4b^2+17ab}}+\frac{1}{\sqrt{4b^2+4c^2+17bc}}+\frac{1}{\sqrt{4c^2+4a^2+17ca}}\ge \frac{3}{5}. $

Let $a,b,c\ge 0: ab+bc+ca+abc=4.$ Prove that $$\color{black}{\frac{1}{\sqrt{4a^2+4b^2+17ab}}+\frac{1}{\sqrt{4b^2+4c^2+17bc}}+\frac{1}{\sqrt{4c^2+4a^2+17ca}}\ge \frac{3}{5}. }$$ Equality holds at $a=b=...
Sickness's user avatar
2 votes
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Dual function non-overlapping Group Lasso's penalty function

I'm trying to find the dual function of this function (non-overlapping Group Lasso's penalty function): $$ \mathfrak{h}: \mathbb{R}^p \to [0,\infty], \ a \mapsto \sum_{j=1}^{k} \left\| a_{\mathscr{A}...
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Packing problem in the 3D cube - generalisation of the Loomis-Whitney / Finner / Brascamp–Lieb / Holder inequality

I have the following geometrical problem: OBSERVATION Consider the unit cube in 3 dimensions, and N orthogonal parallelepiped of size 1/N x 1/N x 1 (hence each of volume $V=1/N^2$). It is easy to ...
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How to prove $\frac{1}{\sqrt{3a^2+13b}}+\frac{1}{\sqrt{3b^2+13c}}+\frac{1}{\sqrt{3c^2+13a}} \ge \frac{3}{4}$?

Let $a,b,c\ge 0: ab+bc+ca>0$ and $a+b+c=3.$ Prove that $$\frac{1}{\sqrt{3a^2+13b}}+\frac{1}{\sqrt{3b^2+13c}}+\frac{1}{\sqrt{3c^2+13a}} \ge \frac{3}{4}. \tag{*}$$ Here is what I've done so far. By ...
Sickness's user avatar
1 vote
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How to prove $\frac{1}{\sqrt{14a^2+b^2+c^2}}+\frac{1}{\sqrt{14b^2+a^2+c^2}}+\frac{1}{\sqrt{14c^2+b^2+a^2}}\ge \frac{9}{4(a+b+c)}$?

Prove that$$\frac{1}{\sqrt{14a^2+b^2+c^2}}+\frac{1}{\sqrt{14b^2+a^2+c^2}}+\frac{1}{\sqrt{14c^2+b^2+a^2}}\ge \frac{9}{4(a+b+c)},$$holds for all $a,b,c>0.$ I tried to use C-S $\dfrac{1}{x}+\dfrac{1}{...
Sickness's user avatar
1 vote
1 answer
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Lebesgue spaces $\mathscr{L}^{p}$ for $p<1$ [duplicate]

I have this following task for Lebesgue spaces $\mathscr{L}^{p}$ with $p<1$: For $p\in(0,1)$ give an example of a measure space $(X,\mathscr{A},\mu)$ and $f,g\in\mathscr{L}^{p}(X,\mathscr{A},\mu;\...
Lukas Kretschmann's user avatar
1 vote
1 answer
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Prove $\sqrt[3]{3a+bc}+\sqrt[3]{3b+ca}+\sqrt[3]{3c+ab}\le \frac{3}{2}\sqrt[3]{a^2+b^2+c^2+29}$ for $ab+bc+ca=3.$

If $a,b,c\ge 0: ab+bc+ca=3$ then prove $$\sqrt[3]{3a+bc}+\sqrt[3]{3b+ca}+\sqrt[3]{3c+ab}\le \frac{3}{2}\sqrt[3]{a^2+b^2+c^2+29}.$$ I tried to use AM-GM, Cauchy-Schwarz without success. Notice that $a^...
Dragon boy's user avatar
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1 answer
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Finding $\small{\min\limits_{a+b+c=1}P=\sqrt{2a^3+abc}+\sqrt{2b^3+abc}+\sqrt{2c^3+abc}. }$

Let $a,b,c\ge 0: a+b+c=1.$ Find the minimum of $P$ $$\color{black} {P=\sqrt{2a^3+abc}+\sqrt{2b^3+abc}+\sqrt{2c^3+abc}. }$$ By $a=b=c=1/3$ we get $P=1.$ Thus, I used AM-GM inequality $$a^3+a^3+abc\ge 3\...
Sickness's user avatar
-1 votes
2 answers
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Prove $\color{black}{\sqrt{\frac{7a+2}{b+c}}+\sqrt{\frac{7b+2}{a+c}}+\sqrt{\frac{7c+2}{b+a}}\ge \frac{9\sqrt{2}}{2} },$ if $a+b+c+abc=4.$

Problem. Let $a,b,c\ge 0: ab+bc+ca>0$ and $a+b+c+abc=4.$ Prove that $$\color{black}{\sqrt{\frac{7a+2}{b+c}}+\sqrt{\frac{7b+2}{a+c}}+\sqrt{\frac{7c+2}{b+a}}\ge \frac{9\sqrt{2}}{2} .}$$ Source: ...
Sickness's user avatar
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upper bound over multiplication of two expectation

Let $f(x)$ be a non-negative function. Is there an upper bound for the following quantity? $$\mathbb{E}\left[f(x)^{1-ab} \right] \mathbb{E}\left[f(x)^{a} \right]^b \leq \, ??,$$ where $a > 0$ and $...
pmoi's user avatar
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Prove $\color{black}{\frac{1}{\sqrt{a+b+2}}+\frac{1}{\sqrt{c+b+2}}+\frac{1}{\sqrt{a+c+2}}\ge1+\frac{1}{\sqrt{2(a+b+c-1)}}.}$

Question. Let $a,b,c\ge 0: a+b+c+abc=4.$ Prove that$$\color{black}{\frac{1}{\sqrt{a+b+2}}+\frac{1}{\sqrt{c+b+2}}+\frac{1}{\sqrt{a+c+2}}\ge1+\frac{1}{\sqrt{2(a+b+c-1)}}.}$$ Because equality holds at $a=...
Dragon boy's user avatar
2 votes
5 answers
108 views

prove that $\frac{1}{a(b + c)^{3}} + \frac{1}{b(c + a)^{3}} + \frac{1}{c(a + b)^{3}} \ge \frac{27}{8}$ given $a^{2} + b^{2} + c^{2} = 1$

\begin{align} \text{Given } a, b, c > 0 \text{ with } a^{2} + b^{2} + c^{2} = 1, \text{ prove that} \\ \frac{1}{a(b + c)^{3}} + \frac{1}{b(c + a)^{3}} + \frac{1}{c(a + b)^{3}} \ge \frac{27}{8}. \...
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If $f \in L_1(0,1)$, then $f^p \log^q (f) \in L_1(0,1)$.

I was wondering if somebody could help me this homework question. Find all $p, q > 0$ with the following properties: (a) if $0 \leq f \in L_1(0, 1)$, then $f^p \log^q(f) \in L_1(0, 1)$. This is for ...
banana_free's user avatar
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1 answer
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Hölder's inequality for functions of multiple variables

I have one question about the Hölder's inequality. We know that the functions in the continuous Holder inequality are f(x) and g(x). If the function were changed to f (x, y) and g (x, y) would the ...
jiawei gao's user avatar
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3 answers
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Prove $\sqrt{\frac{a+b}{c+ab}}+\sqrt{\frac{c+b}{a+cb}}+\sqrt{\frac{a+c}{b+ac}}\ge 3,$ if $ab+bc+ca=3.$

If $a,b,c>0: ab+bc+ca=3,$ then prove that$$\sqrt{\frac{a+b}{c+ab}}+\sqrt{\frac{c+b}{a+cb}}+\sqrt{\frac{a+c}{b+ac}}\ge 3.$$ I've tried to use AM-GM without success. Indeed, we need to prove $$(a+b)(...
Dragon boy's user avatar
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3 answers
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If $a$ and $b$ are positive real numbers such that $a^{2015}+b^{2015}=2015$, find the maximum value of $a+b$.

Hint: Just use the power mean inequality $$\left(\frac{a^{2015}+b^{2015}}{2}\right)^{1/2015}\geq\frac{a+b}{2}$$ My Attempt I was unable to figure out the solution, using the power mean inequality, but ...
Jessie's user avatar
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2 votes
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Nash Inequality

As an exercise from my book I wanted to prove: For $n>2$ and for any $u \in W_0^{1,2}\left(\mathbb{R}^n\right) \cap L^1\left(\mathbb{R}^n\right)$, $$ \int_{\mathbb{R}^n}|\nabla u|^2 d x \geq c\...
Algebraix's user avatar
1 vote
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Brazilian Mathematical Olympiad 2023, Level U, Problem 3

Question Prove that there exists a constant $C > 0$ such that, for any integers $m, n$ with $n \geq m > 1$ and any real number $x > 1$,$$\sum_{k=m}^{n}\sqrt[k]{x} \leq C\bigg(\frac{m^2 \cdot \...
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Proving Inequality Involving Sums and Square Roots with Given Conditions

Question $$\text{Let } b_{i}\wedge a_{i}>0 \text{ where } i\in{1,2,3,\ldots,n} \nonumber , \sum_{i=1}^{n}(b_{i}) = \lambda \text{ then Prove that} \nonumber \frac{\lambda-(b_{1}+b_{2})}{(b_{1}+b_{...
Martin.s's user avatar
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If $f$ is Holder of order $\alpha$, then $f'$ is holder of order $\alpha-1$

Let $D\subset\mathbb{R}$. We say that $f:D\to \mathbb{R}$ is Holder of order $\alpha>1$ if for the largest integer $l<\alpha$ (i.e. $\alpha-l\in (0,1)$: $f$ is $l$ times differentiable, and ...
mathematica's user avatar
2 votes
1 answer
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Seeking Clarification on an Inequality Involving Roots and Products

Question If $a_j\gt 0$, $j=1,\ldots,m$ and $x_i\ge 0$ $i=1,\ldots,n.\;$ then prove that \begin{align} \sqrt[m]{\prod_{j=1}^{m}\left[a_j+\frac{1}{n}\sum_{i=1}^{n}x_i\right]}&\ge\frac{1}{n}\sum_{i=1}...
Martin.s's user avatar
1 vote
0 answers
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How to verify ihe Interpolation inequality for the weighted Bessel potential spaces?

I am trying to prove the following: Let $w$ be an admissible weight, $p_1,p_2\in[1,\infty)$, $\alpha_1,\alpha_2\in\mathbb{R}$, $\theta\in(0,1)$ and \begin{equation} \alpha=\theta\,\alpha_1+(1-\theta)\...
Aban's user avatar
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1 answer
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Holder inequality for 3 functions

Suppose $f \in L^{\infty}(\Omega)$, $g \in L^2(\Omega)$, and $\Omega \subset \mathbb{R}^n$ is bounded domain. Is the following a correct application for Holder inequality? $$ \int_{\Omega} f^2 \cdot ...
Rudinberry's user avatar
1 vote
1 answer
156 views

Prove $\frac{a}{\sqrt{5-4bc}} + \frac{b}{\sqrt{5-4ac}} + \frac{c}{\sqrt{5-4ab}} \ge 1$ With the condition $a+b+c=3$

$$\frac{a}{\sqrt{5-4bc}} + \frac{b}{\sqrt{5-4ac}} + \frac{c}{\sqrt{5-4ab}} \ge 1$$ With the condition $a+b+c=3$ I am thinking about some type of Cauchy-Schwarz inequality here, but not quite sure how ...
FabDust's user avatar
  • 163
4 votes
2 answers
437 views

How to prove $\sqrt{\frac{24a+13}{24a+13bc}}+\sqrt{\frac{24b+13}{24b+13ca}}+\sqrt{\frac{24c+13}{24c+13ab}}\ge 3$

Problem. Let $a,b,c\ge 0: ab+bc+ca>0$ where $a+b+c=3.$ To prove that: $$\sqrt{\frac{24a+13}{24a+13bc}}+\sqrt{\frac{24b+13}{24b+13ca}}+\sqrt{\frac{24c+13}{24c+13ab}}\ge 3.$$ I've tried to use AM-GM, ...
TATA box's user avatar
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Homogeneous Inequality problem unsolved

Let ${x,y,z}$ be positive real numbers. Prove that $\sum\limits_{cyc}\frac{x^2}{yz+\frac{x^4}{y^2}+\frac{x^4}{z^2}} \leq 1$ Here’s my try: By A.M.-G.M. we know that $\frac{yz}{2}+\frac{yz}{2}+\frac{x^...
UWU11's user avatar
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2 votes
1 answer
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Why if p-th variation is finite, then q-th variation is also finite for q>p?

Consider function $L^p(X)$ and $L^q(X)$ space with $p<q$ where $X$ is a finite measure space. Then from Holder, I have $L^q(X)$ embedding into $L^p(X)$. In particular, if $\|f\|_q<\infty$, then $...
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0 votes
2 answers
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How to prove $\frac{1}{a^3+c^3}+\frac{1}{b^3+c^3}+2\left[a+b-c(1-\sqrt{3})\right]\ge3\left(\frac{1}{\sqrt[3]{a+c}}+\frac{1}{\sqrt[3]{b+c}}\right).$?

Let $a,b\ge\dfrac{c}{2}\ge0: (c+a)(c+b)>0$. Prove that: $$\frac{1}{a^3+c^3}+\frac{1}{b^3+c^3}+2\left[a+b-c(1-\sqrt{3})\right]\ge3\left(\frac{1}{\sqrt[3]{a+c}}+\frac{1}{\sqrt[3]{b+c}}\right).$$ I ...
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