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Show that if $f$ is Riemann integrable on $[a,b]$, then

$$\left(\int_{a}^{b}f(x)\sin x\ dx\right)^2+\left(\int_{a}^{b}f(x)\cos x\ dx\right)^2\le(b-a)\int_{a}^{b}f^2(x)\ dx.$$

I know I need to use the Hölder Inequality in some form, but I am not sure how to rewrite the left side of the equation. I know I can rewrite the right side as $\left(\int_{a}^{b}\sin^2x+\cos^2x\ dx\right)\left(\int_{a}^{b}f^2(x)\ dx\right)$, but I'm struggling to invoke any established inequalities because of the addition. Thanks for any shared insights!

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  • $\begingroup$ What's $\sin(x)^{2}+\cos(x)^{2}$ equal to? $\endgroup$ Jan 21, 2019 at 16:13
  • $\begingroup$ The direction of the inequality should be reversed. It almost immdiately follows from Caychy-Schwarz. $\endgroup$ Jan 21, 2019 at 16:14

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By C-S $$(b-a)\int\limits_a^bf(x)^2dx=\int\limits_a^bf(x)^2\sin^2xdx\int_a^b1dx+\int\limits_a^bf(x)^2\cos^2xdx\int_a^b1dx\geq$$ $$\geq\left(\int\limits_a^bf(x)\sin{x}dx\right)^2+\left(\int\limits_a^bf(x)\cos{x}dx\right)^2.$$ Actually, $f^2(x)=f(f(x))$ and $f(x)^2=\left(f(x)\right)^2.$

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1.Prove that $<f,g>:=\int_a^bf(x)g(x)dx$ is an inner product.

2.Observe that the norm it induces is $\|f\|=\big{(}\int_a^bf^2(x)dx\big{)}^{1/2}$.

3.Rewrite your inequality as $<f,\sin>^2+<f,\cos>^2\leq(b-a)\cdot\|f\|^2$

4.Using Cauchy-Schwarz, show that $<f,\sin>^2+<f,\cos>^2 \leq \|f\|^2(\|\sin\|^2+\|\cos\|^2)$

5.Observe that $\|\sin\|^2+\|\cos\|^2=\int_a^b\sin(x)^2dx+\int_a^b\cos(x)^2dx=\int_a^b1dx=b-a$.

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$$\left(\int_{a}^{b}f(x)\sin x\ dx\right)^2+\left(\int_{a}^{b}f(x)\cos x\ dx\right)^2=\Big|\int^b_a e^{ix} f(x)\,dx\Big|^2$$

An application of Cauchy-Bunyakovsky's inequality yields $$\Big|\int^b_a e^{ix} f(x)\,dx\Big|\leq\Big(\int^b_a|e^{2ix}|^2\,dx\Big)^{1/2}\Big(\int^b_a|f(x)|^2\,dx\Big)^{1/2}=(b-a)^{1/2}\int^b_a|f(x)|^2\,dx$$

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