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# Integral inequality of a function

Let be an integer and let be a continuous function. Suppose that for all . Show that

Solution

We begin by a lemma:

Lemma: For an integer the   Hilbert matrix is defined by where

It is known that is invertible and if then .

Since , the   Hilbert matrix ,  is invertible there exist real numbers such that

So the polynomial satisfies the conditions

Clearly   is the sum of all the entries of and so . Now let be a real-valued continuous function on such that

Let be the above polynomial.Then since

integrating gives:

and the result follows.

The exercise along with its solution was taken from here.

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