Youngs inequality is a nice inequality which we are using in various concept of mathematics. A visual proof that p ab ris lebesgue measurable and 1. Combining this elementary observation with the hausdor. Youngs inequality is a special case of the weighted amgm inequality. The proof below applies the convexity of logarithm to judiciously chosen inputs. 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. 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.
In 1912, english mathematician william henry young published the highly intuitive in equality, which is later named as youngs inequality. Pushing the result to infinite sequences does not require any clever idea, and is left to the reader no offense. 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. Our proof is simpler and gives youngs inequality and its converse altogether. In case 1 proof of the sharp form of young s inequality for convolutions, first proved by beckner be and brascamplieb brli. An overview of available proofs and a complete proof of youngs. A third geomteric proof can be found on a separate page. Our proof is simpler and gives young s inequality and its converse altogether.
Further improvements of young inequality springerlink. The poincare inequalities in this lecture we introduce two inequalities relating the integral of a function to the integral of its gradient. The following result allows us to treat different powers of a, b. 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. According to barthe, we rewrite the inequality slightly. Let be a measure space and be the complexvalued integrable functions on define the norm of by.
Pdf in this note we offer two short proofs of youngs inequality and prove its reverse. Some of its applications are envisaged for the development of proofs of other theorems and results. William henry young, english mathematician 18631942 hausdorffyoung inequality, bounding the coefficient of fourier series. If a, b are nonnegative numbers and 1 p inequality can be proven by elementary means when b 6 0 let x ap. The latter also proved a sharp reverse inequality in the case of exponents less than 1. Young s convolution inequality, bounding the convolution product of two functions. It is very useful in real analysis, including as a tool to prove holders inequality. Youngs inequality for products can be used to prove holders inequality. Recap 3 good ways to prove a functional inequality. If a, b are nonnegative numbers and 1 p youngs inequalities. It is also a special case of a more general inequality known as young s inequality for increasing functions. It is also a special case of a more general inequality known as youngs inequality for increasing functions. Applying youngs convolution inequality yields the desired estimate.
Finally, we give new inequalities which are extensions and improvements for the inequalities shown by dragomir. The proof can be easily obtained by induction on n. William henry young, english mathematician 18631942 hausdorff young inequality, bounding the coefficient of fourier series. Youngs inequality and its generalizations introduction. Sharp inequalities in harmonic analysis summer school, kopp august 30th september 4th, 2015 organizers. Operator iteration on the young inequality journal of.
The inherent inequality a s t b t sp1 ab extra a s t b t sp1 ab extra since f2 lp. 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. Pdf an extension of youngs inequality flaviacorina. Proof if either x 0or y 0the result is trivially true. Apparently youngs inequality actually is a special case of the weighted amgm inequality. In this note we offer two short proofs of youngs inequality and prove its reverse. They are the dirichletpoincare and the neumannpoincare in equalities. The inequality is named after william henry young and should not be confused with youngs convolution inequality. In a recent paper, youngs inequality has been seen in a di erent light by.
The nondecreasing sequence two puzzle whose most elegant solution relies on youngs inequality. As in previous arguments, there exist positive numbers h anda. Pdf youngs inequality in traceclass operators martin. It is so straightforward that its proof is often omitted. A proof of holders inequality noncommutative analysis. One can show that equality in the sharp young inequality implies that the functions have to be gaussian. Mar 22, 2012 we coomplete the proof of the young inequality. Youngs convolution inequality, bounding the convolution product of two functions. Pdf a simple proof of the holder and the minkowski inequality. H olders inequality for x and yin rn, xn i1 jx iy ij jjjj pjjxyjj q where pand qare dual indices. 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 proof we are going to present applies to 3 proof. Young inequality also called the babenkobeckner inequality.
Youngs inequality an overview sciencedirect topics. The heart of the matter is to prove the inequality for finite sequences. The proof was postponed, because it uses the holders inequality and the proof of holders inequality will use the youngs inequality. Our proof is based on the arithmeticgeometric mean inequality and will shorten considerably the lines of proof given by y. The most familiar form of young s inequality, which is frequently used to prove the wellknown h. We give elementary proof for known results by dragomir, and we give remarkable notes and some comparisons. If a, b are nonnegative numbers and 1 p inequality m k above is a considerable re nement to the wellknown young inequality and has many nice applications. Proof 1 i have lifted from a discussion at mathoverflow.
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 simple proof of the holder and the minkowski inequality. Youngs inequality for convolution and its applications in. Norms a norm is a function that measures the lengths of vectors in a vector space. Young s inequality is a special case of the weighted amgm inequality. Click here for a proof as mentioned in the introduction, young s inequality is essential in the proof of holders inequality. We have used holders inequality and triangular inequality. The rst thing to note is young s inequality is a farreaching generalization of cauchys inequality. 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. Using the young inequality, we can show that proposition. The present paper characterises the cases of equality in this young inequality, and the characterisation is examined in the.
The classical convolution inequality of young asserts that for all func tions f. If a and b are traceclass operators, and if u is a partial isometry, then, where 1 denotes the norm in the trace class. Nov 18, 2017 we focus on the improvements for young inequality. A convenient proof of young s inequality was given in by utilizing the convexity of the antiderivative function f of equation 1. In section 3, utilizing the refined young inequality and iteration method, we establish some weighted arithmeticgeometric mean inequality for two positive operators. The interpolation of youngs inequality using dyadics.
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. The proof was postponed, because it uses the holders inequality and the proof of holders inequality will use the young s inequality. If there were no proof of it here, we would have a horrible example of circular reasoning in mathematics. 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. A convenient proof of youngs inequality was given in by utilizing the convexity of the antiderivative function f of equation 1. The proof combines perturbative techniques with the sharpened version of the linear hausdorff young inequality due to christ. A version of youngs inequality for convolution is introduced and employed to some topics in convex and setvalued analysis. Note that the p q 2 case has an even simpler proof.
Proof for two random variables, the formula holder inequality 3 may be rewritten as since we integrate with. In rn, we use the cauchyschwarz inequality or simply the schwarz inequality, which states that for all v,w. A visual proof that p ab jul 23, 2018 minkowski inequality proof this video is about the proof of one of the important inequalities named as minkowski inequality. The aim of this note is to give a new proof of the inequality m k.
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