Prove the inequality: $$\left(1+\dfrac{1}{\sin a}\right)\left(1+\dfrac{1}{\cos a}\right)\ge 3+2\sqrt{2}; \text{ for } a\in\left]0,\frac{\pi}{2}\right[$$

This question already has an answer here: Solving an inequality : $n \geq 3$ , $n^{n} \lt (n!)^{2}$. [duplicate] 1 answer

In Hardy’s Pure Mathematics it says if $x^2<2$, $ \ \ y^2>2$, $ \ \ 2-x^2 < \delta$ and $y^2 – 2 < \delta$, then $y-x<\delta$. I added the last two inequalities to get $(y+x)(y-x)<2\delta$. How do I proceed from here?

Let $a_i,b_i,c_i$ be $>0$ ($1 \leq i \leq n$). Then we have $$ \sum (a_i+b_i+c_i) \sum \frac{a_i b_i + b_i c_i + c_i a_i}{a_i+b_i+c_i} \sum \frac{a_i b_i c_i}{a_i b_i+b_i c_i + c_i a_i} \leq \sum a_i \sum b_i \sum c_i $$ This is problem #68 in Hardy, Polya and Littlewood’s Inequality. It can be proved […]

Is it true that $c_m+c_n$ $>$ $c_{m+n}$ for all $m$, $n$ $\in$ $\mathbb{N}$? Though the result seems true, I can’t get a solution. Even the bounds on $c_n$ obtained from Prime Number Theorem isn’t helping me. Is there any way to prove it?

In another post an inequality referred to as “Etemadi’s Inequality” is mentioned twice – in the original post as well as in the answer. However, the contexts of usage are such as to raise the question whether the inequality intended by the users (ziT and saz, respectively) is the inequality that goes by the same […]

Part of the Schwarz-Pick Theorem states that for an analytic automorphism of the unit disk, then $$ \frac{|f'(z)|}{1+|f(z)|^2}\leq\frac{1}{1-|z|^2}. $$ In the wikipedia article of the Schwarz-Pick theorem, it is mentioned that if equality holds, then $f$ is a Moebius transformation on the unit disk without proof. Is there a proof of this detail? Thank you.

$A, B, C$ are the angles of a triangle then $tan^2(A/2)+tan^2(B/2)+tan^2(C/2)$ is always greater than what integral value.

As the title says.. it says to use the mean value theorem but I don’t see how that’s applicable. Thank you

Let $x,y,z>0$ and such $xy+yz+xz\ge 2(x+y+z)$,show that $$\dfrac{1}{xy+z}+\dfrac{1}{yz+x}+\dfrac{1}{zx+y}\le\dfrac{1}{2}$$

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