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Questions tagged [karamata-inequality]

Inequalities, which we can prove by the Karamata's inequal;ity

7
votes
4answers
240 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 ...
6
votes
2answers
662 views

If a,b,c are sides of a triangle, prove: $ \sqrt{a+b-c} + \sqrt{b+c-a} + \sqrt{c+a-b} \le \sqrt{a} + \sqrt{b} + \sqrt{c} $

I did substitute $a=x+y, b=x+z, c=y+z$ and I arrived at $\sqrt{2x} + \sqrt{2y} + \sqrt{2z} \le \sqrt{x+y} + \sqrt{x+z} + \sqrt{y+z}$. However, after this, I tried various methods like AM-GM and Cauchy-...
6
votes
1answer
146 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}\...
5
votes
4answers
404 views

How to show $(a + b)^n \leq a^n + b^n$, where $a, b \geq 0$ and $n \in (0, 1]$?

Does anyone happen to know a nice way to show that $(a+b)^n \le a^n+b^n$, where $a,b\geq 0$ and $n \in (0,1]$? I figured integrating might help, but I've been unable to pull my argument full circle. ...
5
votes
3answers
162 views

How prove this inequality

in $\Delta ABC$, if $A,B,C\in (0,\pi/2]$,show that $$\sin{A}+\sin{B}+\sin{C}>2$$ This problem have many nice methods? Thank you
5
votes
4answers
144 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 ...
5
votes
3answers
126 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)...
4
votes
4answers
230 views

Showing that $ 1<\sin\frac{\alpha}{2}+\sin\frac{\beta}{2}+\sin\frac{\gamma}{2}$

I would like to show that: $$ 1<\sin\frac{\alpha}{2}+\sin\frac{\beta}{2}+\sin\frac{\gamma}{2}$$ where $\alpha, \beta, \gamma$ are the angles of a triangle. I know that the inequality $$ 1<\...
3
votes
3answers
165 views

How do I go about solving this equation? $3^x + 10^x = 4^x + 9^x$

How do I go about solving this equation? $$3^x + 10^x = 4^x + 9^x.$$ I noticed that $1$ and $0$ are solutions, so maybe a way to prove that they are the unique solutions. Taking the derivative does ...
3
votes
6answers
209 views

Is this inequality true? $ (x + y)^{\alpha} < x^{\alpha} + y^{\alpha} $, for positive $x$ & $y$, and for $0 < \alpha < 1$

If $0 < \alpha < 1$, then $$ (x + y)^{\alpha} < x^{\alpha} + y^{\alpha} $$ for $x$, $y$ positive. Is this inequality true in general? I tried using Young's Inequality: For $z,t &...
2
votes
3answers
142 views

If $a^2+b^2+c^2+d^2=4$ what is the range of $a^3+b^3+c^3+d^3$?

I tried to used AM-GM but could not do.Setting any one as $2$ or $-2$ and the others $0$ we get the range as $[-8,8]$ but what is the formal way to do this?
2
votes
5answers
461 views

Maximum of $x^3+y^3+z^3$ with $x+y+z=3$

It is given that, $x+y+z=3\quad 0\le x, y, z \le 2$ and we are to maximise $x^3+y^3+z^3$. My attempt : if we define $f(x, y, z) =x^3+y^3 +z^3$ with $x+y+z=3$ it can be shown that, $f(x+z, y, 0)-f(...
2
votes
3answers
209 views

Maximize $P=a^2+b^2+c^2+ab+ac+bc$

For real numbers $a, b, c$ that satisfy $a + b + c = 6$ and $0\leq a,b,c \leq 4$, maximize $P=a^2+b^2+c^2+ab+ac+bc$. My try: $$\begin{align} \begin{cases} a+b+c=6(1) \\ 0\leq a,b,c\leq4(2) \...
2
votes
1answer
132 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 $.
2
votes
2answers
102 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.
2
votes
0answers
41 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
2answers
306 views

Given the positive real numbers $0\le a,b,c\le 2$ and $a+b+c=3$. Prove that $a^3+b^3+c^3\le 9$ [closed]

Given the positive real numbers $$0\le a,b,c\le 2$$ and $$a+b+c=3$$. Prove that $$a^3+b^3+c^3\le 9$$
1
vote
2answers
52 views

Find least upper bound of $\{a^{2018} + b^{2018} + c^{2018} | a + b + c = 1, a, b, c > 0\}$

Find least upper bound of $\{a^{2018} + b^{2018} + c^{2018} | a + b + c = 1, a, b, c > 0 \}$. I tried power mean inequality, but could only find greatest lower bound.
1
vote
3answers
108 views

Prove that positive $x,y,z$ satisfy $\sqrt{x+y}+\sqrt{y+z}+\sqrt{z+x}\ge \sqrt{2x}+\sqrt{2y}+\sqrt{2z}$

Prove that positive $x,y,z$ satisfy $$\sqrt{x+y}+\sqrt{y+z}+\sqrt{z+x}\ge \sqrt{2x}+\sqrt{2y}+\sqrt{2z}$$ Actually, this is a part of my solution to another problem, which is: If $a,b,c$ are sides ...
1
vote
2answers
91 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 ...
1
vote
2answers
40 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
vote
2answers
48 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
2answers
59 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?
1
vote
2answers
74 views

How do I prove this trigonometric inequality? [duplicate]

If $A,B,C \in (0,\frac{\pi}{2})$. Then prove that $$\frac{\sin(A+B+C)}{\sin(A)+\sin(B)+\sin(C)} < 1$$
1
vote
1answer
66 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
25 views

General Method to solve Power Sum Inequality

This is a general method to have power sum inequality . We work with $x_i> 1$ $n$ real numbers . We want to show this kind of inequality : Let $x_i> 1$ be $n$ real positive numbers and $...
1
vote
1answer
76 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 ...
1
vote
0answers
65 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
0answers
67 views

Karamata + Jensen = Niculescu's inequality (version of 1991)

In fact it's my point of view but I think that this version of Niculescu's inequality is a mixture between Karamata and Jensen's inequality . So we have : Let $f(x)$ be a convex strictly ...
0
votes
1answer
59 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, ...
0
votes
1answer
128 views

Upper bound for sum of exponentials

I want to show that for any vector $a\in \mathbb{R}^n$ with $\sum\limits_{i=1}^n a_i = 0$ and $m=\textrm{max}_i |a_i|$, it holds $$\frac{1}{n} \sum\limits_{i=1}^n e^{a_i} \leq \cosh (m)\, .$$ I ...
0
votes
1answer
59 views

Petrovic and Jensen inequaltiy

I have been reading a book on some classic inequalties and i have stumbled upon this: Let $f:[0, +\infty]\rightarrow \mathbf R$ be a convex function and $x_1,x_2,...,x_n$ a sequence of positive ...
0
votes
1answer
49 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 ...
0
votes
1answer
29 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 ?
0
votes
0answers
23 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 ...
0
votes
1answer
49 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&...