lets-f-0-T-R-does-1-T-0-T-f-t-dt-1-T-0-T-f-t-2-dt-1-T-0-T-f-t-dt-

Question Number 4716 by 123456 last updated on 28/Feb/16

lets f : [0, T] \to R

does ?

\frac{1}{T} \underset{0}{\int^{T}} f (t) d t ⩽ \sqrt{\frac{1}{T} \underset{0}{\int^{T}} {[f (t)]}^{2} d t} ⩽ \frac{1}{T} \underset{0}{\int^{T}} ∣ f (t) ∣ d t

Commented by prakash jain last updated on 29/Feb/16

Let us say T = 1

f (t) = {\begin{cases} 2 & 0 < t ⩽ 0.5 \\ 4 & 0.5 < t ⩽ 1 \end{cases}

\frac{1}{T} \underset{0}{\int^{T}} f (t) d t = 1 + 2 = 3

\sqrt{\frac{1}{T} \underset{0}{\int^{T}} {[f (t)]}^{2} d t} = \sqrt{4 \times . 5 + 16 \times . 5} = \sqrt{10} > 3

\frac{1}{T} \underset{0}{\int^{T}} ∣ f (t) ∣ d t = 3 < \sqrt{10}

The first and third part on inequality evalutes

to same value if f (t) ⩾ 0 .

So the given condition is not true .