Hölder's Inequality for Finite Sums/Euclidean Plane

Hölder's Inequality for Sums: Euclidean Plane

In the context of a Euclidean plane, Hölder's Inequality for Sums takes the following form:

$x_1 y_1 + x_2 y_2 \le \paren {\size {x_1}^p + \size {x_2}^p}^{1 / p} \paren {\size {y_1}^q + \size {y_2}^q}^{1 / q}$

where:

$p, q > 1$
$\dfrac 1 p + \dfrac 1 q = 1$


Sources

  • 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Hölder's inequality
  • 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Hölder's inequality
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