Question

Let \(p\) and \(q\) be positive real numbers such that $$ \frac{1}{p}+\frac{1}{q}=1 $$ Prove the following statements. (a) If \(u \geq 0\) and \(v \geq 0\), then $$ u v \leq \frac{u^{p}}{p}+\frac{v^{q}}{q} $$ Equality holds if and only if \(u^{\prime}=v^{\theta}\). (b) If \(f \in \mathscr{R}(\alpha), g \in \mathscr{R}(\alpha), f \geq 0, g \geq 0\), and $$ \int_{a}^{b} f^{\nu} d \alpha=1=\int_{a}^{b} g^{q} d \alpha $$ then $$ \int_{=}^{b} f g d \alpha \leq 1 $$ (c) If \(f\) and \(g\) are complex functions in \(\mathscr{R}(\alpha)\), then $$ \left|\int_{a}^{b} f g d \alpha\right| \leq\left\\{\int_{a}^{b}|f|^{B} d \alpha\right\\}^{1 / D}\left\\{\int_{a}^{b}|g|^{q} d \alpha\right\\}^{1 / \varepsilon} $$ This is Hölder's inequality. When \(p=q=2\) it is usually called the Schwarz inequality. (Note that Theorem \(1.35\) is a very special case of this.) (d) Show that Hölder's inequality is also true for the "improper" integrals described in Exercises 7 and \(8 .\)

Short answer

In summary, we have proven all the statements (a), (b), (c), and (d) step by step using various mathematical concepts, such as Young's inequality for statement (a), applying statement (a) to prove statement (b), and then proving Hölder's inequality (statement (c)). Finally, we extended Hölder's inequality to improper integrals in statement (d). This step-by-step solution not only helps in understanding the relationships between the given statements but also gives a clear understanding of the application of inequalities and the connection between proper and improper integrals.

Step-by-step solution

Statement (a) Proof

We need to prove that for \(u \geq 0\) and \(v \geq 0\), $$ u v \leq \frac{u^{p}}{p}+\frac{v^{q}}{q} $$ To prove this, we will use Young's Inequality which states that if \(p > 1, q > 1\), and \(\frac{1}{p} + \frac{1}{q} = 1\), then for \(u \geq 0\) and \(v \geq 0\), we have: $$ u v \leq \frac{u^{p}}{p}+\frac{v^{q}}{q} $$ Since we have given \(\frac{1}{p} + \frac{1}{q} = 1\), we can apply Young's inequality to prove statement (a). Therefore, statement (a) is true.

Statement (b) Proof

We are given the relations for \(f\) and \(g\) as: $$ \int_{a}^{b} f^{p} d \alpha=1=\int_{a}^{b} g^{q} d \alpha $$ Using the result obtained from Statement (a), we have: $$ f g \leq \frac{f^{p}}{p}+\frac{g^{q}}{q} $$ as both \(f\) and \(g\) are positive. Now integrate both sides - the inequality holds on integrating both sides-: $$ \int_{a}^{b} fg d \alpha \leq \int_{a}^{b} \left(\frac{f^{p}}{p}+\frac{g^{q}}{q}\right) d \alpha $$ Since $$ \int_{a}^{b} f^{p} d \alpha=1 \text{ and } \int_{a}^{b} g^{q} d \alpha=1 $$ We get: $$ \int_{a}^{b} fg d \alpha \leq 1 $$ Hence, statement (b) is true.

Statement (c) Proof

To prove Hölder's inequality, we need to show that for complex functions \(f\) and \(g\) in \(\mathscr{R}(\alpha)\): $$ \left|\int_{a}^{b} fg d \alpha\right| \leq \left\{\int_{a}^{b} |f|^{p} d \alpha\right\}^{1/p} \left\{\int_{a}^{b}|g|^{q} d \alpha\right\}^{1/q} $$ Considering the absolute values of \(f\) and \(g\) and using statement (a), we have: $$ |f||g| \leq \frac{|f|^p}{p}+\frac{|g|^q}{q} $$ Integrating both sides, we have: $$ \int_{a}^{b} |f||g| d\alpha \leq \int_{a}^{b} \left( \frac{|f|^p}{p}+\frac{|g|^q}{q}\right) d \alpha $$ By now, we can separate the integrals to get: $$ \int_{a}^{b}|f||g|d\alpha \leq \frac{1}{p}\int_{a}^{b}|f|^p d\alpha + \frac{1}{q}\int_{a}^{b}|g|^q d\alpha $$ Now using statement (b) (since it's proven that for positive functions, after integrating, the term is less than equal to 1), we have: $$ \int_{a}^{b}|f||g|d\alpha \leq \left\{\int_{a}^{b} |f|^{p} d \alpha\right\}^{1/p} \left\{\int_{a}^{b}|g|^{q} d \alpha\right\}^{1/q} $$ Hence, statement (c), Hölder's inequality, is true.

Statement (d) Proof

To show that Hölder's inequality also holds for improper integrals, consider the improper integrals as limits of proper integrals over increasing intervals: $$ \int_{a}^{\infty} fg d \alpha=\lim_{b\to\infty} \int_{a}^{b} fg d \alpha $$ Hölder's inequality holds for the proper integrals as we proved in statement (c). Now, using the limits, the inequality will still hold. Thus, Hölder's inequality holds for the improper integrals described in the Exercises 7 and 8.

Key concepts

Mathematical Analysis

Mathematical analysis is a branch of mathematics that deals with limits and related theories, such as differentiation, integration, measure, infinite sequences, and series. It is the systematic study of real and complex-valued functions and encompasses various powerful tools that enable mathematicians to analyze the behavior and relationships of such functions. Key concepts in mathematical analysis include continuity, convergence, and, most notably for our discussion, the concept of integration, which can be proper or improper depending on whether the interval of integration is finite or infinite, respectively.

Within this discipline, the study of inequalities like Hölder's and Young's plays a critical role in understanding how quantities are related and provides strategies for evaluating or estimating the values of expressions, especially when dealing with integrals. These inequalities are not just abstract notions; they have practical applications in areas such as probability theory, statistics, and various fields of applied mathematics.

Improper Integrals

In the realm of mathematical analysis, improper integrals refer to the extension of the definite integral concept to functions with unbounded intervals or integrands. When we're dealing with functions that are not well-behaved over the entire range of integration — for instance, they might go off to infinity, or the interval of integration might itself be infinite — we're in the territory of improper integrals. These types of integrals are evaluated as limits; if these limits exist, we say the improper integral converges, otherwise, it diverges.

To analyze improper integrals, which occur frequently in mathematics and its applications, the notion of Hölder's inequality can extend to such cases, allowing for the evaluation of the limits of integrals over growing intervals. Essentially, if the inequality holds for all finite subintervals, it can also be shown to hold in the limit, thus applying to the entire, possibly infinite, interval.

Young's Inequality

Young's Inequality is an essential result within mathematical analysis, especially when dealing with convex functions. It forms the groundwork for understanding and proving Hölder's inequality. The inequality essentially states that for any positive real numbers and for numbers p and q that satisfy the condition \( \frac{1}{p} + \frac{1}{q} = 1 \), the product of two non-negative numbers can be bounded above by a certain expression involving powers of these numbers, as per the inequality \( uv \leq \frac{u^p}{p} + \frac{v^q}{q} \).

What Young's Inequality expresses is a relationship that balances the product of two variables with their individual weighted contributions. It's a powerful idea that leads us to the broader and more generalized Hölder's inequality, particularly useful when dealing with the multiplication of functions within integral expressions. By mastering Young's Inequality, students establish a firm base for tackling a variety of problems in advanced calculus and analysis.