Some comments on the two functions f and g Mansour

Some comments on the two functions f and g Mansour Mahmoud Firstly f (x) =

−1 ∑ 1 + x (nk + x)2 n=1

f (x) =

−1 1 + 2 ψ ′ (x/k + 1). x k

∞

can be rewritten as

Using the integral representations ∫

1 1 = m z (m − 1)!

ψ

(m)

(z) = (−1)

tm−1 e−zt dt,

m∈N

0

∫

and

∞

∞

m+1 0

we get

∫ f (x) =

tm e−zt dt, 1 − e−t

∞

z > 0; m ∈ N

M (t, k)e−xt dt,

0

where M (t, k) = −1 +

t k(ekt

− 1)

,

t > 0; k > 0.

The function M (t, k) will be negative for all t > 0 iff k > 1. Then the function f will be completely monotonic iff k > 1. Remark 1: We can easily see that the function f will be decreasing for k = 1/2. The function

1 g(x) = 2 + f (x) = 2x

where N (t, k) =

∫

∞

N (t, k)e−xt dt,

0

t t −1+ , kt 2 k(e − 1)

t > 0; k > 0.

Remark 2: We can see that g is increasing for k > 1 and decreasing for k < 1.

1