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Padoa's inequality is named after Alessandro Padoa (1868-1937): Let $a$, $b$, $c$ be sidelengths of a given triangle $\triangle ABC$ then $$(b+c-a)(c+a-b)(a+b-c) \le abc .$$ My question: Is …

Is the inequality as follows true? Let $k > 0$, $a_i$ Is a complex number for $1\le i\le n$ and let $$S:=a_1+a_2+....+a_n$$ Suppose that $$b_i:=S-ka_i \quad\text{ for} \quad 1\le i\le n.$ …

Generaliation the result in our paper for sum and similarly my previous question for product. I have a question: My question: Distributing $N$ points on the sphere so that the sum of their mutual dis …

I am looking for a proof of the inequality as follows: Let $A_1A_2....A_n$ be the regular polygon incribed in a circle $(O)$ with radius $R$. Let $B_1B_2....B_n$ be a polygon incribed the cir …

Let $a_1 \ge a_2 \ge \cdots \ge a_n \ge 1$ or $0 \le a_1 \le a_2 \le \cdots \le a_n \le 1$ and $\alpha_1 \ge \alpha_2 \ge \cdots \ge \alpha_n \ge 1$ then $$\left(\sum_{i=1}^{n}{\alpha_i} \right)\lef …

Using my computer I discovered that: if $a,b,c$ are sidelengths of a triangle, then $(a+b-c)^\alpha+(b+c-a)^\alpha+(c+a-b)^\alpha \ge a^\alpha+b^\alpha+c^\alpha $ if $\alpha >1$ $(a+b-c)^\alpha+(b …

Rearrangement inequality: Assume we have finite ordered sequences of nonnegative real numbers $0 \le a_1 \le a_2 \le\cdots\le a_n \quad\text{and}\quad 0\le b_1 \le b_2 \le\cdots\le b_n, \cdots\,, \qua …

Power sum and elementary symmetric polynomial Let $x_1,. . . , x_n$ be variables, denote for $k \ge 1$ by $p_k(x_1,\dots,x_n)$ the $k-th$ power sum: $$ p_k(x_1,\dots,x_n)=\sum\nolimits_{i=1}^nx_i^ …

Could You give a poof, comment or reference for the inequality as follows: $$\sum_{k=1}^n(-1)^k\frac{\sin kx}{k^{\alpha}} < 0$$ for all $n=1,2,3,\ldots$ and $0<x<\pi$ and $\alpha \ge 1$ See also: …

Update: A year ago, but the first answer is not clear with me. I bounty this question again. My question: I am looking for a proof or counterexample to the following inequality: If $n \in \mathb …

I am looking for a proof of the inequality as follow: Let $n$ be an integer number $n \ge 2$ and $x_1, \cdots, x_n$ and $y_1,\cdots, y_n$ are nonegative real numbers such that $(x_1,\cdots, x_n)$ …

$\DeclareMathOperator\Area{Area}\DeclareMathOperator\cotg{cotg}$I am looking for a proof (or a reference) of an inequality related to area and the sidelengths of a polygon as follows: Let $A_1A_2\cdo …

I am looking for a proof, reference, comment of an inequality as follows: If $f(x)$ and $g(x)$ be two continuous derivative funcions in interval $[a, b]$. Such that: $f(a)=g(a)$ and $f( …

Ono's inequality is true for acute triangle but false with general triangles. The inequality as follows is false with general triangls but I think it true with acute triangle (follows answer by Fedor …

Let $C$ is a perimeter of a convex hull (plane geometry) and $d_{max}$ is the largest distance of two arbitrary points in the convex hull. I am looking for a proof that: $$\frac{C}{d_{max}} \le \pi …