Functional analysis of fa – Isoperimetric inequality for analytical functions in a ring

Leave $ f $ be an anylytic function on the disk drive $ | z | <$ 1. It is well known that
$$ left ( int_0 ^ {2 pi} f (e ^ {i theta}) d theta right) ^ 2 geq 4 pi iint_ {| z | <1} | f (re ^ {i theta}) | ^ 2nd dr d theta. $$

I wonder if the constant $ 4 pi $ can be improved in a ring $ a <z ​​<1 $. More precisely, does the following inequality

$$ left ( int_0 ^ {2 pi} f (e ^ {i theta}) d theta + int_0 ^ {2 pi} f (ae ^ {i theta}) d theta right ) ^ 2 geq C iint_ {a <| z | <1} | f (re ^ {i theta}) | ^ 2r dr d theta $$

wait for some constant $ C (a)> 4 pi $ independent of $ f $? They can $ C (a) $ be calculated in terms of $ to $? This seems to be a classic problem, but I couldn't find a reference and couldn't test it after trying it for a couple of days.

mathematical optimization – Iterate Minimize in a list of functions

I have a function $ f $ defined in $ (- 1,1) $.
For a minimal example, it is sufficient to define

f(z_):=z^2 - 1

I need to find a list of points so that $ f (z_0) = f (z_1) $, points whose image "is at the same height".
I continued to find the minimum through

  min = First@Minimize(f(z), {z})

what happens for $ z $ equal to

argmin = Values@Last@Minimize(f(z), {z})

Also I created a list with

  rang = Subdivide(a, 0,10)

spanning the range from minimum to predefined value.

Now I would like to find, for each item on this list, points such that $ f (z_0) = f (z_1) = rang_j $, for each item in the list.

I couldn't find a better plan than defining a list of features $ fun_j = (f (z) + rang_j) ^ 2 $. By changing the original function and squaring, I am sure that the functions $ fun_j $, one for each item in the list $ rang $ they are positive everywhere except the roots.

So I wanted to iterate over the list of functions a restricted minimization through the commands (the argument $ f_j $ just to clarify my question, I understand that the syntax will be different, that's exactly what the question is about):

   Minimize({f_j, z > argmin}, {z})
   Minimize({f_j, z > argmin}, {z})

that is, run two minimizations one on the left and one on the right of the $ {arg , min} $. I know for mathematical reasons that there are two unique solutions.

I create my list of functions as

 f1(z_,c_):=f(z)+c

and then using

 f1(z,rang)

but I find it hard to iterate Minimize, any suggestion would be helpful.

Annoying

  Minimize({f1( z, rang), z > b}, z)

produces an error message, since the Minimize function argument is expected to be a scalar function.
I'd also like to hear about better methods, in general and in reference to Mathematica.
Health

functions: new data framework for different groups

I have a data frame with a & # 39; Country & # 39; column. Each country has multiple records. I want to write a function in Python using pandas so that for each country I can return a separate data frame.

Have a list of countries like, countries = [& # 39; USA, & # 39; Spain & # 39;, & # 39; France & # 39;],

Individually I can do for

df_us = df [df.country == & # 39; US & # 39;]
df_spain = df [df.country == & # 39; Spain & # 39;],

But how can it be done using a function! Thanks in advance

set-return functions in postgres

I've been trying to use Postgres's generate_series () function to get a table like this:

enter the image description here

I still haven't come up with a solution, so any ideas would be appreciated.

Complex variables cv.com: switch arrays of complex functions

Leave $ z_0 in Bbb {R} $ be arbitrary matrices $ A_0: = A (z_0) $ Y $ B_0: = B (z_0) $ are normal; in view of $ A_0 ^ * = A ^ # ( bar {z_0}) = A ^ # (z_0) $ Y $ B_0 ^ * = B ^ # ( bar {z_0}) = B ^ # (z_0) $ they travel with their Hermitian deputies. They also travel with each other. So $ A_0 $ Y $ B_0 $ it could be simultaneously diagonalized by a unitary matrix. That matrix is ​​also diagonalized $ A_0 ^ * $ Y $ B_0 ^ * $. So the four matrices $ A_0 = A (z_0), B_0 = B (z_0), A_0 ^ * = A ^ # (z_0), B_0 ^ * = B ^ # (z_0) $ it could be diagonalized simultaneously, implying that they all travel with each other. So $ B (AA ^ #) = (AA ^ #) B $ at any point on the actual line. Since the inputs are complete functions, they coincide throughout the complex plane.

calculus and analysis: limit on infinity of arbitrary functions

Here is the code that takes the limit of an expression.

Limit((-I E^(I x) f1(y))/(g2^(Prime)(Prime))(y), x -> (Infinity) )

and the output returned is INDETERMINATE while the desired output is $ infty $. Or if I had to do this instead

Limit((g2^(Prime)(Prime))(y)/(-I E^(I x) f1(y)), x -> (Infinity) )

I would like to get 0 and not INDETERMINATE.

How would you tell Mathematica that the $ f $ Y $ g $ What functions are irrelevant when evaluating the limit?

Thanks for any help.

functions: how does following two (same) calculations give two different results?

I have the following two pieces of code that give me two different results,

  N((-Kp tp + Lc - tp Lc)/(Kp tp)) /. {Lc -> 6, tc -> 0.8, tp -> 0.2, Kp -> 1/3}

Which gives the answer as 23 (The correct answer).

(-Kp tp + Lc - tp Lc)/( Kp tp) /. {Lc -> 6, tc -> 0.8, tp -> 0.2, Kp -> 1/3}

Which gives the answer as 3 (which is obviously wrong).

Could someone explain how / why this happens and how to avoid this (possible) error?

Below is the image of the calculation on my machine:
Calculations on my machine

Question about elliptical functions

Can anyone explain the following behavior in Mathematica 12.0?

EllipticK[N[1/2, 100]]

spit ComplexInfinity. Nevertheless

EllipticK[1/2] // N[#, 100] &

It seems to give the correct result. In my current code, I only know the argument of the function EllipticK numerically then the second option is not exactly what i want.

CHAIN ​​REVERSION in PYTHON (without using built-in functions)

What is wrong with the following code?
It is giving an error: ROPE OUT OF REACH

def reverse(s):
    i=0
    l=()

    while i

parametric functions – Problem: solving the system of equations

I am trying to solve a "simple" system of equations for a formal theoretical model that I am developing. Although I think I had correctly specified the system, I can't find a solution so far. Is it a problem with so many parameters? Can someone help me with this?

Solve({A1*A2*y - A3*y - A4 - A5*((1/A6)^(1-A7))*((x*A8 + A9)^(A10)) + A11*((A6)^(-A12))*((y*A13)^(A14)) == 0, A15*A16*x - A17*x -A19 - A18*A6+ A5*((1/A6)^(1-A7))*((x*A8 + A9)^(A10)) - A11*((A6)^(-A12))*((y*A13)^(A14)) == 0, 00, A2>0,A3>0,A4>0,A5>0,A6>0,A7>0,A8>0,A9>0,A10>0,A11>0,A12>0,A13>0,A14>0,A15>0,A16>0,A17>0, A18>0, A19>0}, {x,y})

Thank you!