An Elegant Inequality

TL;DR

The paper introduces and proves a new inequality involving powers of x and 1-x, which is elegant and not classifiable under known inequality types, requiring novel proof techniques.

Contribution

It presents a novel inequality with a unique structure and provides a proof that overcomes transcendental equation challenges, expanding the landscape of inequality research.

Findings

New inequality $(x)^p + (1-x)^{1/p} \\leq 1$ for $p \\geq 1$, $x \\in [0, 1/2]$.

Proof involves novel techniques handling transcendental equations.

Inequality is not categorizable under known types like Hölder or Minkowski.

Abstract

A new inequality, $(x)^{p}+(1-x)^{\frac{1}{p}}\leq1$ for $p \geq 1$ and $\frac{1}{2} \geq x \geq 0$ is found and proved. The inequality looks elegant as it integrates two number pairs ($x$ and $1-x$, $p$ and $\frac{1}{p}$) whose summation and product are one. Its right hand side, $1$, is the strict upper bound of the left hand side. The equality cannot be categorized into any known type of inequalities such as H\"{o}lder, Minkowski etc. In proving it, transcendental equations have been met with, so some novel techniques have been built to get over the difficulty.