Questions tagged [karamata-inequality]

Inequalities, which we can prove by the Karamata's inequality

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2
votes
1answer
59 views

Given 4 numbers $a, b, c, d> 0,$ show $16\max\limits_{\bigcirc}\left \{ a^{3}+ 3bcd \right \}\!\geq\!\left ( a+ b+ c+ d \right )^{3}$

Given four positive numbers $a, b, c, d.$ Prove that $$16\max\left \{ a^{3}+ 3bcd, b^{3}+ 3cda, c^{3}+ 3dab, d^{3}+ 3abc \right \}\geq\left ( a+ b+ c+ d \right )^{3}$$ the way I think is using the ...
4
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2answers
75 views

With $n> 3,$ prove that $a_{1}+ a_{2}+ a_{3}\geq 100$ by using Karamata's inequality

Given $n$ real numbers $a_{1}, a_{2}\cdots a_{n}$ so that $$a_{1}\geq a_{2}\geq\cdots\geq a_{n}, a_{1}+ a_{2}+ \cdots+ a_{n}= 300, a_{1}^{2}+ a_{2}^{2}+ \cdots+ a_{n}^{2}> 10000$$ With $n> 3,$ ...
2
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0answers
63 views

Prove that :$f(a^2)+f(b^2)+f(c^2)\leq f\left(\frac{1}{4}\right)+f\left(\frac{1}{4}\right)+f(0)$ where $f(x)=\sqrt{\frac{1+\sqrt{1+x}}{x^x}}$

Hi it's a problem found by myself : Let $0<x<1$ then define : $$f(x)=\sqrt{\frac{1+\sqrt{1+x}}{x^x}}$$ Then let $a,b,c>0$ such that $a+b+c=1$ then we have : $$f(a^2)+f(b^2)+f(c^2)\leq f\left(\...
2
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1answer
61 views

Given $a,b,x>0$, $x<y$, prove $(a^x+b^x)^{1/x} > (a^y+b^y)^{1/y}$ [duplicate]

I'm thinking about proving $f(x) = (a^x+b^x)^{1/x}$ has negative derivative for all positive $x$. $$f'(x) = \left(b^x+a^x\right)^\frac{1}{x}\left(\frac{b^x\ln\left(b\right)+a^x\ln\left(a\right)}{\left(...
1
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1answer
58 views

Prove : $\sqrt{\frac{a^2+b^2+1}{a^2+c^2+1}}+\sqrt{\frac{a^2+c^2+1}{b^2+c^2+1}}+\sqrt{\frac{b^2+c^2+1}{a^2+b^2+1}}\geq 3$ with $a+b+c=1$

Claim : Let $a\geq b\geq c\geq 0$ and $a+b+c=1$ then we have : $$\sqrt{\frac{a^2+b^2+1}{a^2+c^2+1}}+\sqrt{\frac{a^2+c^2+1}{b^2+c^2+1}}+\sqrt{\frac{b^2+c^2+1}{a^2+b^2+1}}\geq 3$$ To prove it I try ...
1
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0answers
32 views

Intermediate inequality for an inequality from Crux mathematicorum

I propose a little refinement of this an difficult inequality from Crux mathematicorum . Claim: Let $a,b,c>0$ such that $a\geq b\geq c$ and $\frac{a}{c}\geq\frac{b}{a}\geq \frac{c}{b}$ and $b\geq ...
1
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3answers
83 views

Does this inequality hold true for all $\alpha\in\mathbb{R}$?

Let $x$ and $y$ be two real and positive numbers. Let $\alpha\in\mathbb{R}$. I am trying to understand if the inequality $$ x^{\alpha} + y^{\alpha} \leq (x+y)^{\alpha}$$ holds true. By attemps, I ...
1
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1answer
116 views

An inequality associated with sum of n real numbers

Let $x_i\geq 1, 1\leq i\le n.$ Then show that $$\sum_{i=1}^n\frac{1}{1+x_i}\leq \frac{n-1}{2}+\frac{1}{1+x_1x_2\cdots x_n}.$$ One method of proving this is induction on $n.$ This can be achieved by ...
1
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5answers
95 views

Help me prove the exponential inequality $2^{\sin^2 x} + 2^{\cos^2 x} \leq 3$

Please help me with this inequality: $$2^{\sin^2 x} + 2^{\cos^2 x} \leq 3$$ I've reduced it to this: $$2^t + 2^{1-t} \leq 3 \,\,\,\, \text{where}\,\,t=\sin^2 x$$ and I also did a proof that is not ...
-1
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1answer
42 views

If $0 < \alpha < \alpha + \delta < \beta < \frac\pi2$ then $\tan\alpha + \tan\beta > \tan(\alpha + \delta) + \tan(\beta - \delta).$

For reasons that probably don't bear examination (I've rewritten my answer to this question, but I haven't posted the new improved version with added vitamins, because I wish to supplement my ...
7
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1answer
246 views

Why does this Olympiad inequality proving technique (Isolated Fudging) work?

A few years ago, I was in a math olympiad training camp and they taught us a technique to prove inequalities. I just came across it again recently. However, I am not able to understand why it works. ...
2
votes
1answer
130 views

Prove using Jensen's inequality that if $abcd=1$ then $\frac{1}{(1+a)^{2}}+\frac{1}{(1+b)^{2}}+\frac{1}{(1+c)^{2}}+\frac{1}{(1+d)^{2}} \geq 1$

Question - Let $a, b, c, d$ be positive real numbers such that abcd $=1 .$ Prove that $$ \begin{array}{c} \frac{1}{(1+a)^{2}}+\frac{1}{(1+b)^{2}}+\frac{1}{(1+c)^{2}}+\frac{1}{(1+d)^{2}} \geq 1 \\ \...
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3answers
40 views

Proving an inequality involving real powers

Q: For $p\in\mathbb{R}$ such that $2<p<\infty$, and $a,b>0$, prove the following $$a^p+b^p<(a^2+b^2)^{p/2}$$ My attempt: I say it's equivalent to proving that $(a^p+b^p)^{1/p}<(a^2+b^...
1
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1answer
324 views

Prove or Disprove this statement .

I can't find a counter-example to the following statement : Let $a,b,c>0$ such that $a+b+c=1$ and $a\geq b\geq c$ and $13a^2+5b^2\geq 13b^2+5c^2\geq 13c^2+5a^2$ then $\exists n>1$ such ...
0
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0answers
171 views

A special case of Karamata's inequality to solve one or more Olympiad inequality?

I was wondering if we can always use Karamata's inequality in Olympiad problem .For that I use a special case of Karamata's inequality : If $a_1\geq a_2\geq a_3\geq\cdots\geq a_n$ and $b_1\geq ...
1
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4answers
120 views

Find the minimum value of $x$ s.t. $\sqrt{\left(\frac{x+y}{2}\right)^3}+\sqrt{\left(\frac{x-y}{2}\right)^3}=27$

Let $x,y\in \mathbb{R}$ such that $$\sqrt{\left(\frac{x+y}{2}\right)^3}+\sqrt{\left(\frac{x-y}{2}\right)^3}=27$$. Find the minimize value of $x$. [Edit by Michael Rozenberg] I tried to use AM-GM to ...
1
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1answer
49 views

Extrema of a multivariable function with trigonometric functions.

I am trying to find and classify the extrema of the following function: $f(x,y,z)=\sin(x)+\sin(y)+\sin(z)-\sin(x+y+z)$, with $0\leq x \leq \pi, 0\leq y \leq \pi, 0\leq z \leq \pi$. I have found three ...
1
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1answer
94 views

Prove that $\sum_{i=1}^n\frac{\sum_{i'=1}^na_{i'}^p - a_i^p}{\sum_{i'=1}^na_{i'}^q - a_i^q}\le n\cdot\frac{\sum_{i=1}^na_i^p}{\sum_{i=1}^na_i^q}$.

Given positives $$\large a_1, a_2, \cdots, a_{n - 1}, a_n$$ $(n \in \mathbb Z^+, n \ge 3)$. Prove that for all naturals $p$ and $q$ such that $p \ge q$, $$\large \sum_{i = 1}^n\frac{\displaystyle \...
0
votes
1answer
29 views

Inequality with variable expoent

Let $a,b \geq 0$, and $n \in\mathbb{N}$, then if $1<p,q <\infty$ with $\frac{1}{p}+\frac{1}{q}=1$ is valid $(a+b)^n \leq p^{n}a^{n}+q^{n}b^{n}$. We would like to know in place of exponent $n$ ...
3
votes
1answer
98 views

Given three non-negative numbers $x,y,z$ so that $x+y+z=100$ and $x,y,z\notin (\!33,34\!)$. Prove that $xyz\leqq 33\cdot 34\cdot 33$ .

Given three non-negative numbers $x, y, z$ so that $x+ y+ z= 100$ and $x, y, z\notin (\!33, 34\!)$. Prove that $$xyz\leqq 33\cdot 34\cdot 33$$ I want to see another solution that is not as same as the ...
2
votes
2answers
122 views

Prove that $\sqrt{x^{3} + y^{3}+1} + \sqrt{z^{3} + y^{3}+1} + \sqrt{x^{3} + z^{3}+1} \ge 2 + \sqrt{2(x^{3} + y^{3}+z^{3})+1}$

Given $x, y,z \ge 0$. Prove that $$\sqrt{x^{3} + y^{3}+1} + \sqrt{z^{3} + y^{3}+1} + \sqrt{x^{3} + z^{3}+1} \ge 2 + \sqrt{2(x^{3} + y^{3}+z^{3})+1} $$ Attempt Notice that $$(x^{3} + y^{3}+1) + (z^{...
8
votes
2answers
249 views

Minimum and maximum sum of squares given constraints

Say that we know that $$\sum_{i=1}^n x_i = x_1+x_2+...+x_n = 1$$ for some positive integer $n$, with $x_1 \le x_2 \le x_3 \le ... \le x_n$. The values of $x_1$ and $x_n$ are also known. How can the ...
0
votes
3answers
85 views

Range of function $f(x)=\sqrt{4-x^2}+\sqrt{x^2-1}$

Finding range of function $f(x)=\sqrt{4-x^2}+\sqrt{x^2-1}$ What i try $$y^2=\sqrt{4-x^2}+\sqrt{x^2-1}\Rightarrow y^2=3+\sqrt{(4-x^2)(x^2-1)}$$ $$y^2=3+\sqrt{4x^2-x^4-4+x^2}=3+\sqrt{5x^2-x^4-4}$$ ...
2
votes
2answers
167 views

Inequality $\frac{\ln(7a+b)}{7a+b}+\frac{\ln(7b+c)}{7b+c}+\frac{\ln(7c+a)}{7c+a}\leq \frac{3\ln(8\sqrt{3})}{8\sqrt{3}}$

I'm interested by the following problem : Let $a,b,c>0$ such that $abc=a+b+c$ then we have : $$\frac{\ln(7a+b)}{7a+b}+\frac{\ln(7b+c)}{7b+c}+\frac{\ln(7c+a)}{7c+a}\leq \frac{3\ln(8\sqrt{3})}{8\...
2
votes
1answer
53 views

Prove that $(\sum_{i=1}^m (x_{i})^{p})^{q}\ge(\sum_{i=1}^m(x_{i})^{q})^{p}$ for all $x_i\in\mathbb R^+$ and $p,q\in\mathbb R^+$ such that $p \le q$

I'm studying the norm $ℓ_∞$. It leads me to prove the following inequality $$\left( \sum_{i=1}^m (x_{i})^{p} \right)^{q} \ge \left( \sum_{i=1}^m (x_{i})^{q} \right)^{p}, \quad (x_1, \ldots,x_m) \in ...
6
votes
5answers
314 views

Upper Bound for a Sum

Can you help me prove the following inequality: $$ (\sum_{k=1}^na_kb_kc_k)^2 \leq \sum_{k=1}^na_k^2\sum_{k=1}^nb_k^2\sum_{k=1}^nc_k^2 $$ where $a's,b's,c's \in \mathrm{R}$ I tried to use Cauchy's ...
0
votes
2answers
46 views

Is there a way to prove the next inequality without using the binomial series?

I have $\forall a\ge 1, x>0$ and I want to show that: $$(x^a+1)^{1/a} \le (1+x)$$ The only way I see is through the binomial series. Can you suggest other approaches?
0
votes
1answer
50 views

Without majorization how to prove this inequality?

I'm interested by the following problem : Let $x_i>0$ be $n$ real numbers and $y_i>0$ be $n$ real numbers such that : $1)$ $\forall i$ and $\forall j$ indices and $i\neq j$ we have : ...
4
votes
0answers
77 views

Stronger than Jensen's inequality

I'm interested by the following problem : Let $f(x)$ be a twice differentiable function on an interval $I$ with : 1)$f''(x)\geq 0\quad \forall x \in I$ 2)$f(x)\neq \text{constant ...
4
votes
2answers
128 views

Number of real roots of $3^x+4^x=2^x+5^x$ with proof [duplicate]

This equation $$3^x+4^x=2^x+5^x$$ has two obvious real roots. The question is if it has more real roots than two. A proof is required in any case.
0
votes
3answers
79 views

Find the maximum value of $x^3 + y^3 + z^3$ where $x, y, z \in [0, 2]$ and $x + y + z = 3$. [duplicate]

Given that $x, y, z \in [0, 2]$ and $x + y + z = 3$. Calculate the maximum value of $$\large x^3 + y^3 + z^3$$ I'm done. Should you have different solutions, you could post them down below. Having a ...
3
votes
2answers
272 views

Find maximum and minimum of $\sin x + \sin y$

I am working on my scholarship exam practice but I am stuck on finding the minimum. Pre-university maths background is assumed. When $x + y = \frac{2\pi}{3}, x\geq0, y\geq0$, the maximum of $\...
-1
votes
2answers
202 views

New bounds for convex function of 2 variables

It's related to my post : New bound for Am-Gm of 2 variables In fact I have discovered (maybe it's already knew) a new formula for convex function this is the following : Let $f(x)$ be a twice ...
5
votes
3answers
152 views

Let $A, B$ and $C$ be the angles of an acute triangle. Show that: $\sin A+\sin B +\sin C > 2$. [duplicate]

Let $A, B$ and $C$ be the angles of an acute triangle. Show that: $\sin A+\sin B +\sin C > 2$. I started from considering $$\begin{align}\sin A+\sin B+\sin (180^o-A-B) &= \sin A+\sin B+\sin(A+B)...
1
vote
2answers
59 views

Proving an inequality involving absolute values

How can I prove the inequality $\left|x\right|+\left|y\right|+\left|z\right|\le\left|x+y-z\right|+\left|y+z-x\right|+\left|z+x-y\right|$ for all $x, y, z$ being real number. Can I prove this by ...
1
vote
1answer
82 views

Finding a better bound in an inequality [closed]

Consider points $(x,y)$ on the curve $\sqrt{x^2-3x}+\sqrt{y^2-3y}=1$. Prove that for all such pairs: $$x^2+y^2\lt2(x+y)+8.$$ NOTE.- This problem was proposed by two mathematicians, from Romania and ...
2
votes
0answers
46 views

$a^{ab}+b^{bc}+c^{cd}+d^{da}\geq a^{2a^2b^2}+b^{2b^2c^2}+c^{2c^2d^2}+d^{2d^2a^2}$ with some conditions

I'm interested by the following problem : Let $a,b,c,d$ be real positive numbers suc that $a\geq1$,$b\leq 1$ , $c\leq 1$ , $d\leq 1$ with conditions that I precise below then we have : $$a^{ab}+...
1
vote
1answer
87 views

Generalization of Nesbitts's inequality

Let some (fixed) real $k >0$ and positive reals $a,b,c$. Consider the conjecture $$ \left(\frac{a}{b+c}\right)^k +\left(\frac{b}{a+c}\right)^k+\left(\frac{c}{a+b}\right)^k \geq \min \{\frac{3}{2^k} ...
1
vote
0answers
248 views

$N-1$ equal value principle

Let us start with an example, consider the following sum: $$H_n=\sum_{i=1}^{n}\frac{1}{n-1+x_i}$$ With $x_1x_2...x_n=1$, We're willing to show that $H_n \le 1$. To do this we start with the following ...
1
vote
1answer
170 views

Monotone Increasing Concave Function

Given monotone increasing concave function $f(x):\mathcal{R}_{\geq 0} \to \mathcal{R}_{\geq 0}$, Can we say that $$ f(d_1)+f(d_2)-f(d_1+d_2) \leq f(d_3)+f(d_4)-f(d_3+d_4) $$ if $d_1<d_3$ and $d_2&...
6
votes
1answer
205 views

Proving a convexity inequality

Given $f: \mathbb{R} \to \mathbb{R}$ convex, show that: $$ \frac{2}{3}\left(f\left(\frac{x+y}{2}\right) + f\left(\frac{z+y}{2}\right) + f\left(\frac{x+z}{2}\right)\right) \leq f\left(\frac{x+y+z}{3}\...
0
votes
1answer
62 views

Inequality involving an increasing convex function

I am trying to prove/disprove the following statement: Let $x_1 \geq \cdots \geq x_n $, $ y_1 \geq \cdots \geq y_n $ be real numbers satisfying $ x_1 + \cdots + x_k \leq y_1 + \cdots + y_k $ for all $...
6
votes
4answers
167 views

Inequality $(1+x^k)^{k+1}\geq (1+x^{k+1})^k$

Let $k$ be a positive integer and $x$ a positive real number. Prove that $(1+x^k)^{k+1}\geq (1+x^{k+1})^k$. This looks similar to Bernoulli's inequality. If we write $X=x^k$, the inequality is ...
8
votes
4answers
267 views

Find the number of natural solutions of $5^x+7^x+11^x=6^x+8^x+9^x$

Find the number of natural solutions of $5^x+7^x+11^x=6^x+8^x+9^x$ It's easy to see that $x=0$ and $x=1$ are solutions but are these the only one? How do I demonstrate that? I've tried to write them ...
0
votes
1answer
66 views

Looking for an inequality for $1 \leq p < \infty$

Let $a_1,...,a_n$ be positive real numbers and let $0 < p < 1$. Then $$(a_1 + \cdots + a_n)^p \leq a_1^p + \cdots + a_n^p. $$ Now take $ 1 \leq p < \infty$. Can we get a similar inequality, ...
1
vote
2answers
106 views

Find the minimum value of

Let $a,b,c$ be reals satisfying: (i) $a,b,c\ge0$ (ii) $a+b+c=4$ Find the minimum value of the expression $\sqrt {2a+1}$ $+$ $\sqrt {2b+1}$ $+$ $\sqrt {2c+1}$ So I am literally clueless - I know ...
0
votes
1answer
30 views

Whether $(a-k)^p-(a-h)^p\ge (h-k)^p$ for all $a>h>k>0 ,~ p\ge 1$?

Whether $(a-k)^p-(a-h)^p\ge (h-k)^p$ for all $a>h>k>0 , ~p\ge1$ ? I try some simple value, for example $a=3,h=2,k=1, p=2$ and so on. This inequality is right, but how to prove it ?
2
votes
2answers
117 views

Find maximize and minimize of $P=x+y$

For $\{x,y\}\subset\mathbb R$ such that $\sqrt{x+1}+\sqrt{y+1}=\sqrt{2}\left(x+y\right)$ find maximize and minimize of $P=x+y$ I found the maximize but minimize I have no idea. Help me.
1
vote
2answers
76 views

If $n\geq m$ then $(x^m+y^m)^{1/m} \ge (x^n+y^n)^{1/n}$

If $n\geq m$ show that: $(x^m+y^m)^{1/m} \ge (x^n+y^n)^{1/n}$, all numbers being positive real. Obviously since $n\geq m$, for every $a\in \Bbb Z^+$: $a^{1/m}\geq a^{1/n}$, but does it help?
2
votes
1answer
135 views

Somebody help me please. I have a difficult inequality.

Let $ab+bc+ca=1$. Prove that $2 \ge \sqrt{1+a^2} + \sqrt{1+b^2}+\sqrt{1+c^2}-a-b-c \geq \sqrt3 $.