Problem 5.83

Problems and Theorems in Analysis · 6

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Problem 5.83

The number of positive zeros lying in the interior of the region of convergence does not exceed the number of changes of sign of the sequence

a_1, \quad a_1+a_2, \quad a_1+a_2+a_3, \quad \ldots .

(Generalization of 44.) [80.]

Problem 5.83.1

Let $C$ be the number of changes of sign of the finite sequence

a_1, \quad a_1+a_2, \quad a_1+a_2+a_3, \quad \ldots , \quad a_1+a_2+\cdots +a_n,

$\lambda_i$ ( $i=1,2,\ldots ,n$ ) real numbers satisfying

\lambda _1<\lambda _2<\lambda _3<\cdots <\lambda _n,

and

D(x) = a_1 e^{-\lambda _1 x} + a_2 e^{-\lambda _2 x} + \cdots + a_n e^{-\lambda _n x},

where we assume that $D(0)=0$ . Then $D(x)$ has at most $C+1$ real zeros.

(Both the hypothesis and the conclusion are stronger than in 83. Generalization of 44.1.)