# Questions tagged [karamata-inequality]

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

69 questions
Filter by
Sorted by
Tagged with
134 views

### 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 : $$f''(x)\ge 0$$ ...
• 3,403
1 vote
84 views

### 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 ...
93 views

• 3,403
66 views

• 1,645
44 views

• 3,403
1 vote
129 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 ...
• 413
1 vote
162 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 ...
• 710
1 vote
98 views

• 4,202
466 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 ...
• 7,584
91 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 $$y^2=\sqrt{4-x^2}+\sqrt{x^2-1}\Rightarrow y^2=3+\sqrt{(4-x^2)(x^2-1)}$$ $$y^2=3+\sqrt{4x^2-x^4-4+x^2}=3+\sqrt{5x^2-x^4-4}$$ ...
• 5,379
177 views

• 10.2k
346 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 ...
• 1,169
47 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?
59 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 : ...
99 views

• 315
240 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 ...
190 views

1 vote