An Elegant Inequality

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 Hölder, Minkowski etc. In proving it, transcendental equations have been met with, so some novel techniques have been built to get over the difficulty.

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