Hölder’s inequality in Excel Formulas

Hölder’s inequality is a generalization of the Cauchy-Schwarz inequality, which is a well-known result in mathematics that relates the inner product of two vectors to their norms. Hölder’s inequality extends this idea to more general settings, such as functions and integrals, and allows different exponents for the norms.

The basic idea of Hölder’s inequality is that if we have two functions $f$ and $g$ defined on a measure space $S$, then the integral of their product is bounded by the product of their norms, where the norms are defined using some positive exponents $p$ and $q$ that satisfy $\frac{1}{p}+\frac{1}{q}=1$. The measure space $S$ can be any set with a way of assigning sizes to subsets, such as the real line with the usual length, or a probability space with the probability measure.

The inequality can be written as

where the integrals are taken with respect to the measure on $S$. The norms of $f$ and $g$ are denoted by $\|f\|_p$ and $\|g\|_q$, respectively. The inequality is sharp, meaning that it becomes an equality if and only if $|f|^p$ and $|g|^q$ are proportional, that is, there exist some constants $a$ and $b$ such that $|f|^p=a|g|^q$ almost everywhere on $S$.

Hölder’s inequality has many applications in analysis, such as proving the Minkowski inequality, which is the triangle inequality for the norms $\|\cdot\|_p$, and showing that the dual space of $L^p(S)$, the space of functions with finite $p$-norm, is $L^q(S)$, the space of functions with finite $q$-norm, where $p$ and $q$ are Hölder conjugates. Hölder’s inequality can also be used to derive other inequalities, such as the Young inequality and the Jensen inequality.

Hölder’s inequality was first discovered by Leonard James Rogers in 1888, and then independently by Otto Hölder in 1889, who gave a more general proof and developed the concept of convex and concave functions. The inequality is sometimes also attributed to Frigyes Riesz, who generalized it to more than two functions in 1910 and 1928.

Basic Theory:

Hölder’s Inequality is named after the German mathematician Otto H¨older and is expressed as follows:

Here, and are positive real numbers satisfying the condition . The inequality relates the product of two sequences of numbers to the Lp and Lq norms of those sequences.

Procedures in Excel:

1. Input your data: Create two columns for the sequences of numbers (a and b) in Excel.

Example:

a	b
2	3
4	5
1	2
3	4

2. Calculate the Lp and Lq norms: Create cells to calculate the sum of the absolute values raised to the power of and .

Example:

   

   

3. Apply H¨older’s Inequality: Create a cell to calculate the left side of the inequality using the formulas from step 2.

Example:

   

Then, check if .

Real-world Scenario:

Consider a scenario where you have sales data (a) and corresponding marketing expenses (b) for a product over four months. You want to use H¨older’s Inequality to determine if there is a consistent relationship between sales and marketing expenses.

Calculation:

Let and (satisfying ).

Since , H¨older’s Inequality holds for this scenario.

Other Approaches:

1. Graphical Representation: Create a scatter plot in Excel to visually represent the relationship between the two sequences, aiding in a better understanding.
2. Solver Tool: Utilize Excel’s Solver tool to optimize values for and that satisfy the H¨older’s Inequality with equality.