# Questions tagged [karamata-inequality]

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

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### Correct my sketch of proof about the convexity of the "natural" power tower on $[1,\infty)$

Hi I want to show the following fact : Problem : Let $x\geq 1$ and $n\geq 1$ a natural number and define: $$f(x)={}^{2n}x=\underbrace{x^{x^{⋰^{x}}}}_{2n\text { times}}$$ Then we have : $$f''(x)\ge 0$$ ...
1 vote
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### Show that $(x^n+y^n)^{\frac{1}{n}}$ is monotone where $0\leq x<y$

Let $x,y$ such that $0\leq x<y$. I'm having problem in at proving that the sequence $a_{n}=(x^{n}+y^{n})^{\frac{1}{n}}$ is monotone. I tried it using the function $f(z)=(x^z+y^z)^{\frac{1}{z}}$ and ...
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### 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 vote
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### 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 vote
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### 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 ...
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### 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 $$y^2=\sqrt{4-x^2}+\sqrt{x^2-1}\Rightarrow y^2=3+\sqrt{(4-x^2)(x^2-1)}$$ $$y^2=3+\sqrt{4x^2-x^4-4+x^2}=3+\sqrt{5x^2-x^4-4}$$ ...
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### 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 ...
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### 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?
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### 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 : ... 99 views

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### 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 ... 190 views

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 2 answers 63 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 1 answer 86 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.$$NOTE.- This problem was proposed by two mathematicians, from Romania and ... 2 votes 0 answers 54 views ### a^{ab}+b^{bc}+c^{cd}+d^{da}\geq a^{2a^2b^2}+b^{2b^2c^2}+c^{2c^2d^2}+d^{2d^2a^2} with some conditions I'm interested by the following problem : Let a,b,c,d be real positive numbers suc that a\geq1,b\leq 1 , c\leq 1 , d\leq 1 with conditions that I precise below then we have :$$a^{ab}+... 1 vote