Multiattribute acceptance sampling plans - Library(ISI Kolkata ...

Let us now consider the coefficient of m c 2−i in the first series of the right hand side. For
i = 0 the coefficient of m c 2−i = 0. Let for some i, say i = j ≤ c 1 , the coefficient of m c 2−j > 0.
Then,
(ρ 2 /ρ 1 ) j < (c 2!)(c 1 − j)!
(c 1 !)(c 2 − j)!
This implies
i.e.
(ρ 2 /ρ 1 ) j <
[ ] j
(c2 − j + 1)
(c 1 − j + 1)
...(1.2.11)
(ρ 2 /ρ 1 ) < (c 2 − j + 1)
(c 1 − j + 1) < (c 2 − j)
(c 1 − j) . ...(1.2.12)
Also note that in this case,
c 2 !(c 1 − j − 1)!
(c 2 − j − 1)!c 1 ! > [
ρ2
ρ 1
] j
(c 2 − j)
(c 1 − j) > [
ρ2
ρ 1
] j+1
...(1.2.13)
Thus if for i = j ≥ 1; j ≤ c 1 − 1, the coefficient of m c 2−j > 0, then the coefficient of m c 2−j−1
> 0.
Further if for some j of the first series
(ρ 2 /ρ 1 ) j > c 2!.(c 1 − j)!
c 1 !.(c 2 − j)!
then the coefficient of m (c 2−j) < 0 and it follows that
and thus the coefficient of m (c 2−j+1) < 0.
(ρ 2 /ρ 1 ) j−1 > c 2!.(c 1 − j + 1)!
c 1 !.(c 2 − j + 1)!
It therefore follows that if for some j, 1 ≤ j ≤ c 1 − 1, the coefficient of m (c 2−j) < 0 then for
all i the coefficient of m (c 2−i) < 0, 1 ≤ i < j ≤ c 1 .
From the above three observations we may conclude that the equation F (m) = 0 can have
atmost one real positive root. This is therefore true also for H(m).
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