Question

With the special choice $(5.6)$ one finds $(5.5)$ is negative if it is true that for all positive $x,y$ $$x\log x-x\log y-x+y⩾0$$ with equality only for $x=y.$ Prove this so-called Klein's inequality** directly.

Short answer

Klein's inequality is proven by relating it to the logarithmic mean, which is always nonnegative for positive $x$ and $y$, with equality only when $x=y$.

Step-by-step solution

Simplify the Inequality

Rewrite Klein's inequality to show the relationship between the terms involving log: $$\begin{array}{rlr}x\text{(}\log x-\log y\text{)}-x+y& ⩾0\Rightarrow x\log \left(\frac{x}{y}\right)-x+y& ⩾0\end{array}$$This step uses the logarithm property $\log a-\log b=\log \left(\frac{a}{b}\right)$.

Introduce the Logarithmic Mean

We'll use the logarithmic mean which is defined by: $$L(x,y)=\frac{x-y}{\log x-\log y}\text{for}xeqy$$And is equal to one of the variables if $x=y$. Notice if $x>y$, then $L(x,y)>0$.

Relate to the Logarithmic Mean

Let's rewrite the inequality in terms of the logarithmic mean. We have: $$x\log \left(\frac{x}{y}\right)=x-y\cdot \frac{\log x-\log y}{x-y}$$Substitute the logarithmic mean: $$x\log \left(\frac{x}{y}\right)=x-y\cdot \frac{1}{L(x,y)}$$The inequality now becomes: $$x-y\cdot \frac{1}{L(x,y)}-x+y⩾0$$Or: $$y(1-\frac{1}{L(x,y)})⩾0$$

Analyze the Inequality

We know that $y>0$ since both $x$ and $y$ are positive. So, the inequality will hold if $L(x,y)\ge 1$. The logarithmic mean of two distinct positive numbers is greater than or equal to their geometric mean, which is greater than or equal to 1 when the numbers are equal ($L(x,x)=x$) and is greater for any other pair of positive numbers.

Conclusion and the Case of Equality

Since the logarithmic mean meets the conditions, the inequality holds true for all positive values of $x$ and $y$. Equality is achieved if and only if $x=y$, as in this case, $L(x,y)=x=y$, and all terms on the left-hand side of Klein's inequality cancel each other out, yielding zero.

Key concepts

Logarithmic Mean

The logarithmic mean is an intriguing mathematical concept used to compare two positive numbers, and it's especially useful in various branches of mathematics and engineering. It is defined as:
$$L(x,y)=\frac{x-y}{\log x-\log y}\text{for}xeqy$$
and for cases where $x=y$, it is simply $L(x,x)=x$. What's fascinating about the logarithmic means is that it sits somewhere between the arithmetic and geometric means of two numbers, offering a unique kind of average. It has the remarkable property that if you pick any two distinct positive numbers, $x$ and $y$, their logarithmic mean will always be greater or equal to their geometric mean. This property is crucial when working through proofs of inequalities, just like Klein's inequality. For a more intuitive grasp, consider the logarithmic mean as a way to smoothly transition between two numbers, capturing the essence of their scale in relation to one another through the lens of logarithms.
In the context of the Klein's inequality, the logarithmic mean becomes a key player in demonstrating why this inequality holds true, providing an elegant pathway to seeing the relationships between numbers from a logarithmic perspective.

Inequality Proof

In mathematics, inequality proofs are important for establishing the relations between various expressions or numbers. They are the backbone of many mathematical arguments and are used to confirm the validity of assumptions or to demonstrate certain properties. An inequality proof aims to show that under certain conditions, one quantity is always less than, greater than, or equal to another. One approach, commonly found in proofs like that for Klein's inequality, uses a series of algebraic manipulations and known properties of other mathematical functions to rearrange an inequality into a form that reveals its truth.
In the Klein's inequality proof, for instance, we start by rewriting the inequality in terms of recognizable quantities such as the logarithmic mean. By relating our inequality to this mean, we're leveraging its properties to show that the inequality is satisfied. The final part of a proof often involves dealing with any possible cases of equality. Understanding when and why equality occurs is just as important as proving the inequality itself. Through inequality proofs, students learn to build logical arguments, spot critical properties, and consider all possible scenarios for the mathematical relationships they're studying.

Logarithm Properties

Logarithms are one of the foundational tools in mathematics, with unique properties that make them invaluable for solving a wide range of problems, especially those involving exponential relationships. A few key properties are often employed in mathematical proofs:

• The product rule: ${\log }_a(b\cdot c)={\log }_a(b)+{\log }_a(c)$
• The quotient rule: ${\log }_a(\frac{b}{c})={\log }_a(b)-{\log }_a(c)$
• The power rule: ${\log }_a(b^c)=c\cdot {\log }_a(b)$
• Change of base formula: ${\log }_b(a)=\frac{{\log }_c(a)}{{\log }_c(b)}$

These properties simplify complex expressions and allow us to write exponential relationships in more manageable forms. They're especially useful in proofs, such as Klein's inequality, where we handle expressions with logarithms in several variables. Utilizing these properties wisely enables us to rewrite the original inequality in terms that reveal underlying relationships, such as those involving the logarithmic mean. The quotient rule, for instance, was applied to show that $x\log x-x\log y$ can be expressed as $x\log (\frac{x}{y})$ which set the stage for introducing the logarithmic mean into the inequality. Mastering logarithm properties empowers students to tackle a wide array of problems in calculus, algebra, and beyond.