Exercise 5.10. Given . Prove Nesbit's octal equality for the following three terms:
Solution:
Apply Holder's octal corollary to the following two sequences
Then we have:
Since x1 > 0, ∀i=1,2,3 it follows that:
The "=" sign in (1) occurs:
We prove:
Indeed: (2)
So (2) is true, the equal sign occurs⇔ .
From (1), (2) deduce:
The equal sign occurs ⇔ .
(Apply Hölder's octagonal corollary to calculate the distance from a point to a given line.)
Exercise 5.11. Given a line d : by and a point outside that line. Prove that the distance ρ from point M to d is given by the following formula:
Prove that the distance ρ from point M to d is given by the following formula:
Solution:
Suppose N(x,y) is an arbitrary point on d, that is, we have by Applying the consequence of Hölder inequality to the following two sequences: and we get:
Since
Substituting (2) into (1) we have:
The equal sign in (3) occurs if and only if
From (3) and (6) it follows that:
Draw MH perpendicular to d, then according to the property of the perpendicular, we have:
Exercise 5.12. Let n be a natural number. Prove that:
Solution:
Choose two ranges
As a consequence of Holder's octave, we have:
According to Newton's binomial, we have:
Put
we get:
So from (1) has .
Exercise 5.13. Prove that:
Solution:
we have:
Identifying the coefficients of on both sides of the polynomial, we get:
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