Let $$f(x)=\left(1-\frac{2x}{x+c}\right)^{-n}-\left(1+\frac{x}{c}\right)^{2n}$$ and $$g(x)=\left(1-\frac{2x}{c}\right)^{-n}-\left(1-\frac{x}{c}\right)^{-2n}$$ where $c>0$ and $n>0$ are constants. I am wondering if $f(x)\leq g(x)$ for $0\leq x <c/2$ (I am actually interested in small positive $x$). Clearly, this holds with equality when $x=0$ and I think the inequality holds for $0< x <c/2$. This is based on the numerical evaluations as well as […]

Find all real number $n$ such that $|\frac{\sin(nx)}{n\sin(x)}|\le1\forall x\in\mathbb{R}-\{\pi k: k\in\mathbb{Z}\}$.

I wish to find a sequence of strictly positive real numbers $(a_1, a_2, \dots)$, such that $$ \sum_{k = 1}^\infty \frac{a_k}{k} < \infty $$ and such that for all $m, n \in \{1, 2, \dots\}$ with $m \leq \frac{n}{2}$ the following relation holds $$ \frac{\ln(n – m) \times \ln(m)}{\ln(n)} \leq a_m $$ Any help will […]

I am wondering if the following statement is true for $\theta\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ and $x,y\in\mathbb{R}$: $$\tan^{-1}\left(\frac{\sin(\theta)+x}{\cos(\theta)+y}\right)\leq\theta+x\cos(\theta)-y\sin(\theta)+c(x^2+y^2),$$ where $c\geq2$ is a constant (though an answer showing that constant $c$ exists without quantifying what it is would be good enough). I’ve plotted the difference between RHS and LHS for multiple values of $\theta$ and the inequality seems to hold. […]

Let $A=(a_{ij})_{n\times n}$ such $a_{ij}>0$ and $\det(A)>0$. Defining the matrix $B:=(a_{ij}^{\frac{1}{n}})$, show that $\det(B)>0?$. This problem is from my friend, and I have considered sometimes, but I can’t. Thank you

Let $y_1,\ldots y_n$ be positive real numbers satisfying $y_1+\cdots+y_n\geq n$ and $\displaystyle{\frac{1}{y_1}+\cdots+\frac{1}{y_n}\geq n}$. Is it true that $y_1y_2\cdots y_n\geq 1$?

While answering a question, the OP made a follow – up question, that I was not able to answer at that moment. However, I came up with an intriguing (at least to me) question. Let $\mathcal{S_n}(\mathbb R)$ be the class of all the real $n\times n$, symmetric, positive semi – definite matrices, with one entry […]

Let $a,b,c\in\mathbb{R}^+$, prove that $$\sqrt{\frac{a}{b+c}}+\sqrt{\frac{b}{c+a}}+\sqrt{\frac{c}{a+b}}\leq \sqrt{\frac{3}{2}\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)} $$ Hi everyone, I’ve been trying to do this exercise but any method that I tried has failed. First I tried to use $v=(\sqrt{\frac{a}{b+c}},\sqrt{\frac{b}{c+a}},\sqrt{\frac{c}{a+b}})$ and $w=(1,1,1)$ (the most obvious) but didn’t work. Then I did this $$LHS= \frac{\sqrt{a(c+a)(a+b)}+\sqrt{b(b+c)(a+b)}+\sqrt{c(b+c)(c+a)}}{\sqrt{(b+c)(c+a)(a+b)}}\leq \frac{\sqrt{(a(a+b)+b(b+c)+c(c+a))(c+a+a+b+b+c)}}{\sqrt{(b+c)(c+a)(a+b)}} = \frac{\sqrt{2((a+b+c)^2-(ab+bc+ac))(a+b+c)}}{\sqrt{(b+c)(c+a)(a+b)}}$$ Using the fact that $\sqrt{ax}+\sqrt{by}+\sqrt{cz} \leq \sqrt{(a+b+c)(x+y+z)}$ but […]

Following on from an earlier question, and in search of a conceptual insight, I asked myself: Given real numbers $a \ge b \ge c \ge d \ge 0 \tag{1}$ Prove that $a^n – b^n + c^n – d^n \ge (a – b + c – d)^n \text{ for all } \underline{n \in \mathbb{R}} \tag{2}$ First, […]

I’m trying to figure out why $b^n – a^n < (b – a)nb^{n-1}$. Using just algebra, we can calculate $ (b – a)(b^{n-1} + b^{n-2}a + \ldots + ba^{n-2} + a^{n-1}) $ $ = (b^n + b^{n-1}a + \ldots + b^{2}a^{n-2} + ba^{n-1}) – (b^{n-1}a + b^{n-2}a^2 + \ldots + ba^{n-1} + a^{n-1}) $ $ […]

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