Paper 1, Section II, F

Analysis I | Part IA, 2013

Fix a closed interval [a,b][a, b]. For a bounded function ff on [a,b][a, b] and a dissection D\mathcal{D} of [a,b][a, b], how are the lower sum s(f,D)s(f, \mathcal{D}) and upper sum S(f,D)S(f, \mathcal{D}) defined? Show that s(f,D)S(f,D)s(f, \mathcal{D}) \leqslant S(f, \mathcal{D}).

Suppose D\mathcal{D}^{\prime} is a dissection of [a,b][a, b] such that DD\mathcal{D} \subseteq \mathcal{D}^{\prime}. Show that

s(f,D)s(f,D) and S(f,D)S(f,D)s(f, \mathcal{D}) \leqslant s\left(f, \mathcal{D}^{\prime}\right) \text { and } S\left(f, \mathcal{D}^{\prime}\right) \leqslant S(f, \mathcal{D})

By using the above inequalities or otherwise, show that if D1\mathcal{D}_{1} and D2\mathcal{D}_{2} are two dissections of [a,b][a, b] then

s(f,D1)S(f,D2)s\left(f, \mathcal{D}_{1}\right) \leqslant S\left(f, \mathcal{D}_{2}\right)

For a function ff and dissection D={x0,,xn}\mathcal{D}=\left\{x_{0}, \ldots, x_{n}\right\} let

p(f,D)=k=1n[1+(xkxk1)infx[xk1,xk]f(x)]p(f, \mathcal{D})=\prod_{k=1}^{n}\left[1+\left(x_{k}-x_{k-1}\right) \inf _{x \in\left[x_{k-1}, x_{k}\right]} f(x)\right]

If ff is non-negative and Riemann integrable, show that

p(f,D)eabf(x)dx.p(f, \mathcal{D}) \leqslant e^{\int_{a}^{b} f(x) d x} .

[You may use without proof the inequality ett+1e^{t} \geqslant t+1 for all tt.]

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