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I am reading about Holder inequality in the book Introductory functional analysis with applications by Erwin Kreyszig. When proving the Holder inquality the author uses Fig 5 to explain the inequality in (6). Here is an image of the proof enter image description here

Does anyone know how Fig 5. explains the inequality in 6?

Thanks.

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    $\begingroup$ Compare the area of the rectangle with the sum of the shaded areas. $\endgroup$ – Anurag A Nov 14 '16 at 8:07
  • $\begingroup$ @AnuragA, But what is the difference between the two graphs? $\endgroup$ – MrDi Nov 14 '16 at 8:08
  • $\begingroup$ it is showing two possible scenarios $\beta > \alpha^{p-1}$ and $\beta \leq \alpha^{p-1}$. $\endgroup$ – Anurag A Nov 14 '16 at 8:15
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Observe that in Fig 5 (both figures), area of rectangle is $\alpha \beta$. But the area under the curve with respect to $t$ axis and area of the curve with respect to $u$ axis together constitute larger area than the rectangle in both the cases.

If we have given a rectangle, then either the curve will intersect the rectangle at $u=\beta$(Straight line) first or it will intersect the straight line $t=\alpha$ first.

There is also the third case when the curve passes through the meeting point of $u=\beta$ and $t=\alpha$ in which we achieve equality.

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