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Jan
28
comment Order of Double Coset
@Sushil: The natural 1-1 and onto map is $axb\leftrightarrow axbx^{-1}$ for $a\in A$ and $b\in B$.
Jan
6
comment Annoying Polynomial Inequality
@SouvikDey: Please note that: (1) $h(x)=0$ if and only if $p(x)=0$; (2) $p$ is a polynomial.
Dec
27
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.
Dec
25
comment Holomorphic function $\varphi$ with fixed point $z_0$ such that $\varphi'(z_o)=1$ is linear?
@VictorWang: You are welcome.
Dec
25
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 $r$.
Dec
25
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?
Oct
30
awarded  Enlightened
Oct
26
awarded  Yearling
Sep
30
awarded  Explainer
Jun
3
awarded  Nice Answer
Apr
4
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 $-\frac{\pi}{2}$.
Apr
4
answered Vector spaces and intersections
Apr
3
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
Apr
1
awarded  Revival
Mar
25
awarded  Enlightened
Mar
25
awarded  Nice Answer
Mar
19
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
Mar
10
comment A beautiful inequality for convex functions
@Julien: You are welcome!
Mar
10
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.
Mar
10
revised A beautiful inequality for convex functions
reorganized