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In mathematics, the '''Lebedev–Milin inequality''' is any of several inequalities for the coefficients of the exponential of a power series, found by {{harvs|txt|last1=Lebedev|author1-link=Nikolai Andreevich Lebedev|last2=Milin|year=1965}} and {{harvs|txt|first=Isaak Moiseevich|last=Milin|authorlink=Isaak Moiseevich Milin|year=1977}}. It was used in the proof of the [[Bieberbach conjecture]], as it shows that the [[Milin conjecture]] implies the [[Robertson conjecture]].


They state that if


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:<math>\sum_{k\ge 0} \beta_kz^k = \exp\left(\sum_{k\ge 1} \alpha_kz^k\right)</math>
 
for complex numbers β<sub>''k''</sub> and α<sub>''k''</sub>, and ''n'' is a positive integer, then
 
:<math>\sum_{k=0}^{\infty}|\beta_k|^2 \le
\exp\left(\sum_{k=1}^\infty k|\alpha_k|^2\right),</math>
 
:<math>\sum_{k=0}^{n}|\beta_k|^2 \le
(n+1)\exp\left(\frac{1}{n+1}\sum_{m=1}^{n}\sum_{k=1}^m(k|\alpha_k|^2 -1/k)\right),</math>
 
:<math>|\beta_n|^2 \le
\exp\left(\sum_{k=1}^n(k|\alpha_k|^2 -1/k)\right).</math>
 
See also [[exponential formula]] (on exponentiation of power series).
 
==References==
* {{Citation | last1=Conway | first1=John B. | author1-link=John B. Conway | title=Functions of One Complex Variable II | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-94460-9 | year=1995 | oclc=32014394}}
*{{Citation | last1=Korevaar | first1=Jacob | title=Ludwig Bieberbach's conjecture and its proof by [[Louis de Branges]] | jstor=2323021 | mr=856290 | year=1986 | journal=[[American Mathematical Monthly|The American Mathematical Monthly]] | issn=0002-9890 | volume=93 | issue=7 | pages=505–514 | doi=10.2307/2323021}}
*{{Citation | last1=Lebedev | first1=N. A. | last2=Milin | first2=I. M. | title=An inequality | mr=0186793  | year=1965 | journal=Vestnik Leningrad University. Mathematics | issn=0146-924X | volume=20 | issue=19 | pages=157–158}}
*{{Citation | last1=Milin | first1=I. M. | title=Univalent functions and orthonormal systems | publisher=[[American Mathematical Society]] | location=Providence, R.I. | mr=0369684  | year=1977}} (Translation of the 1971 Russian edition)
 
{{DEFAULTSORT:Lebedev-Milin inequality}}
[[Category:Inequalities]]

Revision as of 06:42, 4 February 2014

In mathematics, the Lebedev–Milin inequality is any of several inequalities for the coefficients of the exponential of a power series, found by Template:Harvs and Template:Harvs. It was used in the proof of the Bieberbach conjecture, as it shows that the Milin conjecture implies the Robertson conjecture.

They state that if

k0βkzk=exp(k1αkzk)

for complex numbers βk and αk, and n is a positive integer, then

k=0|βk|2exp(k=1k|αk|2),
k=0n|βk|2(n+1)exp(1n+1m=1nk=1m(k|αk|21/k)),
|βn|2exp(k=1n(k|αk|21/k)).

See also exponential formula (on exponentiation of power series).

References

  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

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    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

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    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

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    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

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    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

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  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

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