Reaction quotient

In mathematics, Young's inequality is either of two inequalities: one about the product of two numbers, and one about the convolution of two functions. They are named after William Henry Young.

Young's inequality for products can be used to prove Hölder's inequality. It is also used widely to estimate the norm of nonlinear terms in PDE theory, since it allows one to estimate a product of two terms by a sum of the same terms raised to a power and scaled.

Young's inequality for products

Standard version for conjugate Hölder exponents

In its standard form, the inequality states that if a and b are nonnegative real numbers and p and q are positive real numbers such that 1/p + 1/q = 1, then

$ab\leq {\frac {a^{p}}{p}}+{\frac {b^{q}}{q}}.$

Equality holds if and only if a^p = b^q. This form of Young's inequality is a special case of the inequality of weighted arithmetic and geometric means and can be used to prove Hölder's inequality.

Proof

The claim is certainly true if a = 0 or b = 0. Therefore, assume a > 0 and b > 0 in the following. Put t = 1/p, and (1 − t) = 1/q. Then since the logarithm function is strictly concave

$\log(ta^{p}+(1-t)b^{q})\geq t\log(a^{p})+(1-t)\log(b^{q})=\log(a)+\log(b)=\log(ab)$

with equality if and only if a^p = b^q. Young's inequality follows by exponentiating.

Elementary case

An elementary case of Young's inequality is the inequality with exponent 2,

$ab\leq {\frac {a^{2}}{2}}+{\frac {b^{2}}{2}},$

which also gives rise to the so-called Young's inequality with ε (valid for every ε > 0), sometimes called the Peter–Paul inequality. This name refers to the fact that tighter control of the second term is achieved at the cost of losing some control of the first term – one must "rob Peter to pay Paul"

$ab\leq {\frac {a^{2}}{2\varepsilon }}+{\frac {\varepsilon b^{2}}{2}}.$

Proof

Young's inequality with exponent 2 is the special case p = q = 2. However, it has a more elementary proof, just observe that

$0\leq (a-b)^{2}=a^{2}+b^{2}-2ab,$

add 2ab to every side and divide by 2.

Young's inequality with ε follows by applying Young's inequality with exponent 2 to

$a'=a/{\sqrt {\varepsilon }},{\text{ }}b'={\sqrt {\varepsilon }}b.$

Standard version for increasing functions

The area of the rectangle a,b can't be larger than sum of the areas under the functions $f$ (red) and $f^{-1}$ (yellow)

For the standard version of the inequality, let f denote a real-valued, continuous and strictly increasing function on [0, c] with c > 0 and f(0) = 0. Let f⁻¹ denote the inverse function of f. Then, for all a ∈ [0, c] and b ∈ [0, f(c)],

$ab\leq \int _{0}^{a}f(x)\,dx+\int _{0}^{b}f^{-1}(x)\,dx$

with equality if and only if b = f(a).

Generalization using Legendre transforms

If f is a convex function and its Legendre transform (convex conjugate) is denoted by g, then

$ab\leq f(a)+g(b).\,$

This follows immediately from the definition of the Legendre transform. This inequality also holds — in the form a ·b ≤ f(a) + g(b) — if f is a convex function taking a vector argument Template:Harv.

Examples

• The Legendre transform of f(a) = a^p/p is g(b) = b^q/q with q such that 1/p + 1/q = 1, and thus Young's inequality for conjugate Hölder exponents mentioned above is a special case.
• The Legendre transform of f(a) = e^a – 1 is g(b) = 1 − b + b ln b, hence ab ≤ e^a − b + b ln b for all non-negative a and b. This estimate is useful in large deviations theory under exponential moment conditions, because b ln b appears in the definition of relative entropy, which is the rate function in Sanov's theorem.

Young's inequality for convolutions

In real analysis, the following result is also called Young's inequality:

Suppose f is in L^p(R^d) and g is in L^q(R^d) and

${\frac {1}{p}}+{\frac {1}{q}}={\frac {1}{r}}+1$

with 1 ≤ p, q, r ≤ ∞. Then

$\|f*g\|_{r}\leq \|f\|_{p}\|g\|_{q}.$

Here the star denotes convolution, L^p is Lebesgue space, and

$\|f\|_{p}={\Bigl (}\int |f(x)|^{p}\,dx{\Bigr )}^{1/p}$

denotes the usual L^p norm.

An example application is that Young's inequality can be used to show that the heat semigroup is a contracting semigroup using the L² norm (i.e. the Weierstrass transform does not enlarge the L² norm).

In case p, q > 1 Young's inequality can be strengthened to a sharp form, viz

$\|f*g\|_{r}\leq c_{p,q}\|f\|_{p}\|g\|_{q}.$

where the constant c_p,q < 1.

See also

• Convex conjugate
• Legendre transformation

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

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  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
• 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.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  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
  
  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.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

External links

• Young's Inequality at PlanetMath
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