Young inequality proof pdf

Our proof is simpler and gives young s inequality and its converse altogether. Youngs inequality yue kwok choy question 1 let f be a realvalued function which is continuously differentiable and strictly increasing on the interval i 0. Rupert frank california institute of technology, pasadena, usa diogo oliveira e silva university of bonn, germany christoph thiele university of bonn, germany supported by hausdor center for mathematics, bonn. In rn, we use the cauchyschwarz inequality or simply the schwarz inequality, which states that for all v,w. In section 3, utilizing the refined young inequality and iteration method, we establish some weighted arithmeticgeometric mean inequality for two positive operators. Pdf youngs inequality in traceclass operators martin. In this note we offer two short proofs of youngs inequality and prove its reverse. Youngs inequality an overview sciencedirect topics. Pdf in this note we offer two short proofs of youngs inequality and prove its reverse. Finally, we give new inequalities which are extensions and improvements for the inequalities shown by dragomir. Apparently youngs inequality actually is a special case of the weighted amgm inequality.

I realized that the explanation of the former second step of proof below was a little bit obscure since, while entirely correct, did not clarify enough why the choice of integrability exponent is not done by guessing. A third geomteric proof can be found on a separate page. Pdf a simple proof of the holder and the minkowski inequality. Proof for two random variables, the formula holder inequality 3 may be rewritten as since we integrate with. A convenient proof of youngs inequality was given in by utilizing the convexity of the antiderivative function f of equation 1. William henry young, english mathematician 18631942 hausdorffyoung inequality, bounding the coefficient of fourier series. Operator iteration on the young inequality journal of.

One can show that equality in the sharp young inequality implies that the functions have to be gaussian. Youngs inequality is a nice inequality which we are using in various concept of mathematics. Recap 3 good ways to prove a functional inequality. In a recent paper, youngs inequality has been seen in a di erent light by.

Young s convolution inequality, bounding the convolution product of two functions. Further improvements of young inequality springerlink. Applying youngs convolution inequality yields the desired estimate. Young inequality also called the babenkobeckner inequality.

An overview of available proofs and a complete proof of youngs. The inherent inequality a s t b t sp1 ab extra a s t b t sp1 ab extra since f2 lp. Nov 18, 2017 we focus on the improvements for young inequality. If there were no proof of it here, we would have a horrible example of circular reasoning in mathematics. A proof of holders inequality noncommutative analysis. The proof was postponed, because it uses the holders inequality and the proof of holders inequality will use the youngs inequality.

Let be a measure space and be the complexvalued integrable functions on define the norm of by. It is also widely used 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. Pdf an extension of youngs inequality flaviacorina. Youngs inequality for convolution and its applications in.

A visual proof that p ab ris lebesgue measurable and 1. The aim of this note is to give a new proof of the inequality m k. William henry young, english mathematician 18631942 hausdorff young inequality, bounding the coefficient of fourier series. The following result allows us to treat different powers of a, b. Sharp inequalities in harmonic analysis summer school, kopp august 30th september 4th, 2015 organizers. Convexity, inequalities, and norms 9 applying the same reasoning using the integral version of jensens inequality gives p q z x fpd 1p x fqd 1q for any l1 function f. We also obtain reverse ratio type and difference type inequalities for positive operators by means of iteration under different conditions in section 4 and section 5, respectively. The nondecreasing sequence two puzzle whose most elegant solution relies on youngs inequality. The present paper characterises the cases of equality in this young inequality, and the characterisation is examined in the. Note that the p q 2 case has an even simpler proof. In case 1 proof of the sharp form of young s inequality for convolutions, first proved by beckner be and brascamplieb brli.

Pushing the result to infinite sequences does not require any clever idea, and is left to the reader no offense. In 1912, english mathematician william henry young published the highly intuitive in equality, which is later named as youngs inequality. As a matter of fact, the proof of the main result in 1 relies on a parametrization of functions which was used in and was suggested by brunns proof of the brunnminkowski inequality. Combining this elementary observation with the hausdor. Young s inequality is a special case of the weighted amgm inequality. Youngs inequality and its generalizations introduction. The poincare inequalities in this lecture we introduce two inequalities relating the integral of a function to the integral of its gradient. The proof combines perturbative techniques with the sharpened version of the linear hausdorff young inequality due to christ. The proof we are going to present applies to 3 proof. Therefore i decided to substitute it by a similar but more direct procedure and put the former step 2 in the notes for a brief proof of their equivalence. The latter also proved a sharp reverse inequality in the case of exponents less than 1. Youngs, minkowskis, and holders inequalities penn math. As in previous arguments, there exist positive numbers h anda. The proof below applies the convexity of logarithm to judiciously chosen inputs.

We give elementary proof for known results by dragomir, and we give remarkable notes and some comparisons. It is very useful in real analysis, including as a tool to prove holders inequality. A convenient proof of young s inequality was given in by utilizing the convexity of the antiderivative function f of equation 1. Norms a norm is a function that measures the lengths of vectors in a vector space. Using the young inequality, we can show that proposition. Our proof is simpler and gives youngs inequality and its converse altogether. Some of its applications are envisaged for the development of proofs of other theorems and results. Mar 22, 2012 we coomplete the proof of the young inequality. It is also a special case of a more general inequality known as young s inequality for increasing functions.

The proof can be easily obtained by induction on n. The proof was postponed, because it uses the holders inequality and the proof of holders inequality will use the young s inequality. It is so straightforward that its proof is often omitted. If a, b are nonnegative numbers and 1 p inequality can be proven by elementary means when b 6 0 let x ap. Youngs inequality and its generalizations 3 remark 2. Proof if either x 0or y 0the result is trivially true. Youngs inequality for products can be used to prove holders inequality. It is also a special case of a more general inequality known as youngs inequality for increasing functions. The rst thing to note is young s inequality is a farreaching generalization of cauchys inequality. They are the dirichletpoincare and the neumannpoincare in equalities. According to barthe, we rewrite the inequality slightly. Youngs inequality, which is a version of the cauchy inequality that lets the power of 2 be replaced by the power of p for any. A simple proof of the holder and the minkowski inequality.

The heart of the matter is to prove the inequality for finite sequences. The interpolation of youngs inequality using dyadics. Youngs inequality is a special case of the weighted amgm inequality. Our proof is based on the arithmeticgeometric mean inequality and will shorten considerably the lines of proof given by y. H olders inequality for x and yin rn, xn i1 jx iy ij jjjj pjjxyjj q where pand qare dual indices.

Click here for a proof as mentioned in the introduction, young s inequality is essential in the proof of holders inequality. Assume further that the following inequality is called the young inequality since both are nonzero, we can divide both side of the above inequality by and obtain the following equivalent form let us define a new variable using the relation we can see that thus the inequality can be rewritten as. A version of youngs inequality for convolution is introduced and employed to some topics in convex and setvalued analysis. The classical convolution inequality of young asserts that for all func tions f. The most familiar form of young s inequality, which is frequently used to prove the wellknown h. We have used holders inequality and triangular inequality. If a and b are traceclass operators, and if u is a partial isometry, then, where 1 denotes the norm in the trace class. Youngs convolution inequality, bounding the convolution product of two functions.