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 Jan28 comment Order of Double Coset @Sushil: The natural 1-1 and onto map is $axb\leftrightarrow axbx^{-1}$ for $a\in A$ and $b\in B$. Jan6 comment Annoying Polynomial Inequality @SouvikDey: Please note that: (1) $h(x)=0$ if and only if $p(x)=0$; (2) $p$ is a polynomial. Dec27 comment How to show that $f'(x)<2f(x)$ @Hans: Sorry, I don't know any other similar question, and after some failed attempts, I have to admit that I cannot explain my motivation of the "by (1)" step better than the post itself suggests. Dec25 comment Holomorphic function $\varphi$ with fixed point $z_0$ such that $\varphi'(z_o)=1$ is linear? @VictorWang: You are welcome. Dec25 comment Holomorphic function $\varphi$ with fixed point $z_0$ such that $\varphi'(z_o)=1$ is linear? @VictorWang: Sorry, I just read your comment because it is very long time since the last time I visited this website. Please note that since $\varphi(\Omega)\subset \Omega$, $\varphi^k$ is a well-defined holomorphic function on $\Omega$ for every $k$. Then from the choice of $r$ we know the radius of convergence of the Taylor expansion of $\varphi^k$ around $z_0$ is no less than $r$. Dec25 comment How to show that $f'(x)<2f(x)$ @Hans: Thank you for your upvote. I think the last time I visited this website is a couple of months ago, and I just read your comment. I am not sure how to explain the motivation of the step mentioned in your comment clearly. Are you still interested in this question? Oct30 awarded Enlightened Oct26 awarded Yearling Sep30 awarded Explainer Jun3 awarded Nice Answer Apr4 comment Vector spaces and intersections @BeniBogosel: Thank you. Maybe I should mention that the example is a special case of the following observation. By choosing the basis $\{e_1,\dots, e_4\}$ appropriately, $V_1$, $V_2$, $V_3$ can always be of the form given in my answer and at the same time, $V_4$ is spanned by a pair of vectors $(e_1,e_3)A+(e_2,e_4)B$, where $A$, $B$ and $A-B$ are invertible $2\times 2$ matrices. Then $W$ exists iff $A^{-1}B$ has a real eigenvalue. In my answer, $A$ is identity and $B$ is rotation by $-\frac{\pi}{2}$. Apr4 answered Vector spaces and intersections Apr3 revised What is $\sum_{n=0}^{\infty}|a_nz^n|^2=\frac{1}{2 \pi}\int_{-\pi}^{\pi}|f(ze^{it})|^2dt$ for? corrected typo Apr1 awarded Revival Mar25 awarded Enlightened Mar25 awarded Nice Answer Mar19 reviewed Approve Function $f$ from $[0,\infty)$ such that is limit at infinity equals zero and it's values greater than zero must decrease somewhere Mar10 comment A beautiful inequality for convex functions @Julien: You are welcome! Mar10 comment A beautiful inequality for convex functions @Julien: I edited my answer by reorganizing the argument. I hope it looks clearer now. In the current form, your question asks how I get $h_t$ in $(1)$. First I assume $h_t(x)=b\cdot V(x-a)$ for some constants $a$, $b$. Then from $h_t\ge f_t$ we know $b(t-a)=h_t(t)\ge f_t(t)=1$. To get $\int_0^1h_t \le 2\int_0^1f_t=2(1-t)$, I use a stronger inequality $\frac{b(1-a)^2}{2}=\int_a^1 h_t\le 2(1-t)$ instead. The inequalities determine a unique pair of $(a,b)=(2t-1,\frac{1}{1-t})$. There is probably more geometrical explanation, but maybe it needs more words due to the limitation of my English. Mar10 revised A beautiful inequality for convex functions reorganized