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In [[mathematics]], the '''Borel–Carathéodory theorem''' in [[complex analysis]] shows that an [[analytic function]] may be [[bounded]] by its [[real part]]. It is an application of the [[maximum modulus principle]]. It is named for [[Émile Borel]] and [[Constantin Carathéodory]].
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== Statement of the theorem ==
 
Let a function <math>f</math> be analytic on a [[closed disc]] of [[radius]] ''R'' centered at the [[origin (mathematics)|origin]]. Suppose that ''r'' < ''R''. Then, we have the following inequality:
 
:<math> \|f\|_r \le \frac{2r}{R-r} \sup_{|z| \le R} \operatorname{Re} f(z) + \frac{R+r}{R-r} |f(0)|. </math>
 
Here, the norm on the left-hand side denotes the maximum value of ''f'' in the closed disc:
 
:<math> \|f\|_r = \max_{|z| \le r} |f(z)| =  \max_{|z| = r} |f(z)|</math>
 
(where the last equality is due to the maximum modulus principle).
 
== Proof ==
 
Define ''A'' by
 
: <math> A = \sup_{|z| \le R} \operatorname{Re} f(z). </math>
 
First let ''f''(0) = 0. Since ''Re f'' is harmonic, we may take ''A''>0. ''f'' maps into the half-plane ''P'' to the left of the ''x''=''A'' line. Roughly, our goal is to map this half-plane to a disk, apply [[Schwarz's lemma]] there, and tease out the stated inequality.
 
<math>w \mapsto w/A - 1</math> sends ''P'' to the standard left half-plane. <math>w \mapsto R(w+1)/(w-1)</math> sends the left half-plane to the circle of radius ''R'' centered at the origin. The composite, which maps 0 to 0, is the desired map:
 
:<math>w \mapsto \frac{Rw}{w-2A}.</math>
 
From Schwarz's lemma applied to the composite of this map and ''f'', we have
 
:<math>\frac{|Rf(z)|}{|f(z)-2A|} \leq |z|.</math>
 
Take |''z''| &le; ''r''. The above becomes
 
:<math>R|f(z)| \leq r|f(z) - 2A| \leq r|f(z)| + 2Ar</math>
 
so
 
:<math>|f(z)| \leq \frac{2Ar}{R-r}</math>,
 
as claimed. In the general case, we may apply the above to ''f''(''z'')-''f''(0):
 
:<math>
\begin{align}
  |f(z)|-|f(0)|
    &\leq |f(z)-f(0)|
    \leq \frac{2r}{R-r} \sup_{|w| \leq R} \operatorname{Re}(f(w) - f(0)) \\
    &\leq \frac{2r}{R-r} \left(\sup_{|w| \leq R} \operatorname{Re} f(w) + |f(0)|\right),
\end{align}</math>
 
which, when rearranged, gives the claim.
 
== References ==
* Lang, Serge (1999). ''Complex Analysis'' (4th ed.). New York: Springer-Verlag, Inc. ISBN 0-387-98592-1.
* Titchmarsh, E. C. (1938). ''The theory of functions.'' Oxford University Press.
 
{{DEFAULTSORT:Borel-Caratheodory theorem}}
[[Category:Theorems in complex analysis]]
[[Category:Articles containing proofs]]

Latest revision as of 02:22, 2 September 2014

Whether or not you're in the property market in search of residential, industrial, industrial or mixeddevelopment properties. Proper here at LNP, we've got all the new property launches which covers the entire of Singapore. Each of the property launch listed below are nicely documented with complete and up-to-date info & particulars.

The property is meticulously maintained all through and includes substantial rooms with fitted storage cupboards, fully built-in up to date kitchen, mild and ethereal main bedroom with fitted storage, good-sized and totally related for top pace web/cellphone access and fashionable toilet with jacuzzi bathtub You can too add Hamilton Lopez, who claims to be a marine biologist at sea with no phone and wants to buy a home for her/his dad by way of pay pal. This individual can not spell and ask for reassurance they will not be disenchanted. Spends loads of time writing lengthy letters in responce to houses on the market. wanted to hire the condominium.I wasted my money and time.Since then I have determined made a mistake of sending greater than the asking price of the merchandise, this

Know where the bargains are. You probably know some ways to observe for deals on airfare and hotels. Be sure you're also on every possible checklist for printable coupons and low cost codes by signing up for alerts from all your most popular airlines, resorts and rental automobile agencies and following them on Facebook and Twitter. All the time replace them when your property or email address changes. Should you bid for offers on Priceline, give yourself a competitive edge by consulting BiddingforTravel or BetterBidding to seek out the lowest acceptable bids.

c) SSD at 5% if the property search singapore is bought within the third 12 months of buy, i.e. the property is held for more than two years and as much as three years from the date of buy. c. MAS will even decrease the LTV ratio for residential property loans to non-individual debtors (i.e. property holding companies or any company purchase) from 50% to 40%. 7. eleven Jan 2013 – Extra Measures To Guarantee A Stable And Sustainable Property Market – MAS Link b. Impose Additional Buyer Stamp Duty (ABSD) on Everlasting residents buying their 1st residential property and Singaporeans buying their 2nd residential property. iii) Developers of future EC sale websites from the Authorities Land Gross sales programme will solely Centro Residences Rental & Sale Units OBTAINABLE! Bedok Reservoir Condo @ Bedok

Singapore is also rated as a high place in the world for dwelling, working and taking part in. It is globally related, has multiple cultures living there, and is a superb surroundings with many industries and businesses. There is a strong authorized system, properties are close to where you're employed or to leisure, and it has been mentioned by the International Competitiveness Report that Singapore is ranked as one of the most aggressive Asia nations on the earth and is in the prime three in foreign commerce and investment.

A concession to defer Stamp Responsibility fee on all contracts was launched in June 1998 as part of the off-finances measures to cushion the affect of economic slowdown. The Government has decided to withdraw the concession with rapid impact (from 15 December 2006) because the economic conditions and the property market have improved. As a part of the minimum 10% downpayment by the borrower, the minimum cash component for purchases of personal residential properties has additionally been lowered from 10% to five%. In tandem, the Government has decided to reduce the cash downpayment for HDB flats financed with bank loans to 5%, as an alternative of the initially slated 10percent1 on 1 January 2006. This may align the cash downpayment for each HDB and private property.

The place you're buying property which is underneath development, the Singapore Academyof Regulation will maintain a portion of the purchase monies as stakeholders till the expiryof the defects legal responsibility period offered in the Agreement/Contract. GST will not be payable on residential property at present. Nonetheless, as some residentialproperties (e.g. shophouse with a flat above) could involve a commercial componentyou ought to seek recommendation from your lawyer on this. Your Lawyer's Role in a Property Sale Singapore residents buying their second properties will probably be hit with an ABSD of 7% while folks with everlasting residency standing will pay an additional stamp obligation of 5% on their first residence purchase. For more news and analysis on Singapore Budget 2013, click here for ST's Huge Story protection.

Interest rates are extraordinarily low, globally and in Singapore, and proceed so as to add gasoline to our property market. Now we have to take this additional round of measures now, to verify latest market trends and avoid a extra critical correction in prices additional down the road,ђ said Deputy Prime Minister and Minister for Finance Tharman Shanmugaratnam. The TDSR is principally aimed toward encouraging financial prudence among debtors and guaranteeing consistent lending practices among the banks. Banks are actually more stringent of their credit score checking and assessment when granting property loans. After they apply for a loan, borrowers will need to present more documents in addition to full more paperwork. The approval course of can even take an extended time. Real-time market knowledge.

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