if-x-gt-0-r-pq-1-p-q-then-1-rx-1-qx-p-1-px-q-1-x-r-

Question Number 144095 by mathdanisur last updated on 21/Jun/21

i f x > 0, r = p q, 1 ⩽ p ⩽ q t h e n :

1 + r x ⩽ {(1 + q x)}^{p} ⩽ {(1 + p x)}^{q} ⩽ {(1 + x)}^{r}

Answered by mitica last updated on 21/Jun/21

{(1 + x)}^{r} = {[{(1 + x)}^{p}]}^{q} \overset{b e r n o u l l i}{⩾} {(1 + p x)}^{q}

{(1 + q x)}^{p} \overset{b e r n o u l l i}{⩾} 1 + p q x = 1 + r x

f (t) = \frac{l n (1 + t)}{t}

p x ⩽ q x \Rightarrow \frac{l n (1 + p x)}{p} ⩾ \frac{l n (1 + q x)}{q} \Rightarrow {(1 + p x)}^{q} ⩾ {(1 + q x)}^{p}

Commented by mathdanisur last updated on 21/Jun/21

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