NishMath

Recent research has focused on the Boppana entropy inequality, a mathematical relationship that connects the entropy of a squared variable, denoted as H(x²), to the entropy of the variable itself, H(x). This inequality, expressed as H(x²) ≥ φxH(x), where φ is the golden ratio (approximately 1.618), has gained attention for its surprising tightness. Specifically, the maximum error between the two sides of the inequality is less than 2% for large values of x within the range [0,1] and even lower for small values of x.

The Boppana inequality’s significance also extends to coding theory, where it can be rephrased as a statement about the possibility of compressing data with different biases. Some experts have expressed hope for an intuitive information-theoretic or combinatorial proof of this inequality. Furthermore, explorations into the function G(x²)=bxG(x) have shown a connection between the Boppana inequality and the function Ĥ(x), which was found to have surprising symmetry around x = ½.


References :

• math.stackexchange.com: https://math.stackexchange.com/questions/5020685/intuition-for-boppana-entropy-inequality-hx2-geq-x-phi-hx
• mathoverflow.net: Intuition for Boppana Entropy Inequality $H(x^2) \geq x \phi H(x)$
• youtu.be: Intuition for Boppana Entropy Inequality H(x^2) ≥ x φ H(x)

Classification:

• HashTags: #Entropy #InformationTheory #MathAnalysis
• Company: MathOverflow
• Target: Bounds
• Product: Coding
• Feature: EntropyInequality
• Type: Research
• Severity: Medium