Exam

For any

a > 0

take

φ (t) = (t - a)^{+}

. What inequality can be deduced? Find an optimal value for

a

and show that

((\int f^{*} d μ)^{1 / 2} - \sqrt{μ (S) / e})^{2} \leq \int f \log^{+} f d μ + μ (S) / e .

φ^{'} = I_{(a, \infty)}

Φ (t) = \log^{+} (t / a)

\int f^{*} d μ \leq \int (f^{*} - a)^{+} d μ + a μ (S) \leq \int \log^{+} (f^{*} / a) f d μ + a μ (S) \leq \int f \log^{+} f + (a e)^{- 1} f^{*} d μ + a μ (S)

Now

inf_{a > 0} A / a + a B = 2 \sqrt{A B}

and thus for

I : = \int f^{*} d μ

and

A = I / e

B = μ (S)

(\sqrt{I} - \sqrt{μ (S) / e})^{2} = I - 2 \sqrt{I μ (S) / e} + μ (S) / e \leq \int f \log^{+} f d μ + μ (S) / e

It follows that

I \leq (\sqrt{\int f \log^{+} f d μ + μ (S) / e} + \sqrt{μ (S) / e})^{2} \leq 2 \int f \log^{+} f d μ + 4 μ (S) / e .