We introduce the rational functions
$$\begin{aligned}& f_{n} (\beta; x) = \frac{r_{n}}{p_{n}},\quad n=1,2, \dots, \qquad f_{1} \equiv 1, \end{aligned}$$
(3.1)
\(r_{n}\) and
\(p_{n}\) defined by (
2.6). Note that
\(f_{n+1} < f_{n}\) because
\(\{ u_{\nu }/ v_{\nu }\} = \{ 1 / \nu \}\) is a decreasing sequence [
3, p. 10, Problem 28]. The approximating functions
\(F_{n}\) in (
2.17) can now be written as
$$\begin{aligned}& F_{n} = \frac{xf_{n} + \beta }{(x+2) f_{n} + \beta },\quad n=1,2,\dots, \qquad F_{n}(0) = \frac{x}{x+2}. \end{aligned}$$
(3.2)
The
β-derivative of
\(F_{n}\) is
$$\begin{aligned}& F'_{n} = \frac{2}{[(x+2)f_{n} + \beta ]^{2}} \bigl(f_{n} - \beta f'_{n}\bigr), \quad n=1,2, \dots , \end{aligned}$$
(3.3)
and
$$F'_{n}(0) = \frac{2(2n+1)}{3(x+2)^{2}} > 0, \quad n=1,2,\dots . $$
This immediately shows that
\(F'_{n} > 0\) for small positive values of
β. We want to show that
\(F'_{n} > 0\) for
\(\beta \in (0, \infty)\). But first we must establish certain facts about the rational functions
\(f_{n} = r_{n} / p_{n}\). By (
2.5) and with
\(\rho_{\nu }= ( \nu \beta + x - 1)(\nu \beta + x)(\nu \beta + x + 1) (\nu \beta + x +2)\) the constituent terms of
\(r_{n}\) and
\(p_{n}\) are products of
\(n-1\) Hurwitz polynomials, each of degree 4, so that each of those terms is a Huwitz polynomial of degree
\(4(n-1)\). The
β-derivative of
\(f_{n}\) is
$$\begin{aligned} f'_{n} =& \frac{1}{p_{n}^{2}} \bigl( r'_{n} p_{n} - r_{n} p'_{n} \bigr) \end{aligned}$$
(3.4a)
$$\begin{aligned} =& \Biggl[ \Biggl( \sum_{\nu =1}^{n} u'_{\nu } \Biggr) \Biggl( \sum_{\nu =1}^{n} v_{\nu } \Biggr) - \Biggl( \sum_{\nu =1} ^{n} u_{\nu } \Biggr) \Biggl( \sum _{\nu =1}^{n} v'_{\nu } \Biggr) \Biggr] \Biggl( \sum_{\nu =1}^{n} v_{\nu } \Biggr) ^{-2}. \end{aligned}$$
(3.4b)
Using (
2.8) and (
2.9) in (
3.4b), we arrive at
$$\begin{aligned}& \Biggl( \sum_{\nu =1}^{n} v_{\nu } \Biggr) ^{2} f'_{n} = \sum _{1 \leq \mu < \nu \leq n}^{n} \frac{\mu \nu (\nu - \mu)}{\rho_{\mu }^{2} \rho_{\nu }^{2}} \bigl( \rho_{\mu }\rho '_{\nu }- \rho '_{\mu } \rho_{\nu } \bigr) . \end{aligned}$$
(3.5)
Here
$$\begin{aligned} \rho_{\mu }\rho '_{\nu }- \rho '_{\mu } \rho_{\nu }= {}& (\nu - \mu) \bigl\{ \mu^{3} \nu^{3} \alpha_{1} \beta^{6} + 2\mu^{2} \nu^{2} (\mu + \nu) \alpha_{2} \beta^{5} \\ &{}+ \bigl[ 3\mu \nu \bigl(\mu^{2} + \mu \nu + \nu^{2}\bigr) \alpha_{3} + \mu^{2} \nu ^{2} \alpha_{1} \alpha_{2} \bigr] \beta^{4} \\ &{}+ \bigl[ 4\bigl(\mu^{2} + \nu^{2}\bigr) (\mu + \nu) \alpha_{4} + 2\mu \nu ( \mu + \nu) \alpha_{1} \alpha_{3} \bigr] \beta^{3} \\ &{}+ \bigl[ 3\bigl(\mu^{2} + \mu \nu + \nu^{2}\bigr) \alpha_{1} \alpha_{4} + \mu \nu \alpha_{2} \alpha_{3}\bigr] \beta^{2} \\ &{}+ 2(\mu + \nu) \alpha_{2} \alpha_{4} \beta + \alpha_{3} \alpha _{4} \bigr\} > 0, \quad \beta \in (0, \infty), x \in (1, \infty), 1 \leq \mu < \nu \leq n, \end{aligned}$$
the positive functions
\(\alpha_{\nu }\) (
\(\nu = 1,\dots, 4\)) being defined in connection with (
1.12). Thus, by (
3.5),
\(f'_{n} > 0\),
\(\beta > 0\),
\(x>1\). This establishes strict monotonicity of the functions
\(f_{n}\) defined in (
3.1). Note that
\(0 < f_{n} < 1\),
\(n \geq 2\),
\(\beta > 0\) and
$$\begin{aligned}& \begin{aligned} & f_{n} \downarrow \frac{3}{2n+1} \quad \text{as } \beta \downarrow 0, \qquad f_{n} \uparrow \frac{1 + 1/2^{3} + \cdots + 1/n^{3}}{1 + 1/2^{2} + \cdots + 1/n^{2}} = \frac{d_{n}}{e_{n}} \quad \text{as } \beta \uparrow \infty , \\ &f_{n} (\infty) = \lim_{\beta \uparrow \infty } f_{n} \downarrow \zeta (3) / \zeta (2) = 0.730 763\dots \quad \text{as } n \uparrow \infty . \end{aligned} \end{aligned}$$
(3.6)
We also note the following facts about
\(f'_{n}\), which follow from (
3.4a),
$$\begin{aligned}& f'_{n} \uparrow \frac{9\alpha_{3} (n^{2} + n -2)}{\alpha_{4} (2n+1)^{2}} > 0 \quad \text{as } \beta \downarrow 0,\qquad f'_{n} \downarrow 0\quad \text{as } \beta \uparrow \infty . \end{aligned}$$
(3.7)
Furthermore, (
2.5) together with (
1.12) and with the notation used in (
3.6) shows that
$$\begin{aligned}& r_{n} = (n!)^{4} d_{n} \beta^{4(n-1)} + \text{lower order terms}, \\& p_{n} = (n!)^{4} e_{n} \beta^{4(n-1)} + \text{lower order terms}, \end{aligned}$$
so that
$$\begin{aligned}& r'_{n} = 4(n-1) (n!)^{4} d_{n} \beta^{(4(n-1)-1} + \cdots, \\& p'_{n} = 4(n-1) (n!)^{4} e_{n} \beta^{(4(n-1)-1} + \cdots . \end{aligned}$$
Consequently,
$$r'_{n} p_{n} - r_{n} p'_{n} = c_{n} \beta^{8(n-1)-2} + \text{lower order terms}, $$
where
\(c_{n}\) is a positive constant. Since
$$p_{n}^{2} = (n!)^{8} e_{n}^{2} \beta^{8(n-1)} + \text{lower order terms}, $$
we see that
\(f'_{n}\) as given in (
3.4a) behaves like
\(\beta^{-2}\) as
\(\beta \uparrow \infty \). In other words,
\(f'_{n}\) and
\(\beta^{-2}\) are asymptotically proportional. Consequently,
$$\begin{aligned}& \beta f'_{n} \downarrow 0 \quad \text{as } \beta \uparrow \infty . \end{aligned}$$
(3.8)
We now remember the fact that the polynomials
\(\rho_{\nu }\) are Hurwitzian and that, consequently, the constituent terms of the polynomials
\(r_{n}\) and
\(p_{n}\) in (
2.5) are Hurwitzian. By Theorem IV of [
4] (in conjunction with the specification of terminology concerning
circular regions and
circles on p. 164 of [
4]) the sum of any two of these constituent polynomials of degree
\(4(n-1)\) is Hurwitzian. Thus,
\(r_{n}\) and
\(p_{n}\) are Hurwitz polynomials. Their zeros are located in the open left-hand half of the complex
β-plane, which we denote by
L. By another theorem [
5, p. 115], all zeros of
\(f'_{n}\) (and all its poles) are located in
L. In other words,
\(f'_{n} > 0\) for real
\(\beta > 0\), a fact which has been established earlier already by direct means. Applying the theorem of [
5] again, this time to the rational function
\(f'_{n}\), we arrive at the result that
\(f''_{n}\) has all its zeros (and poles) in
L, i.e.,
\(f''_{n} \neq 0\) for real
\(\beta > 0\). Limit relation (
3.7) shows that
\(f'_{n}\) decreases somewhere in the interval
\((0, \infty)\). Thus, the
β-derivative
\(f''_{n}\) of
\(f'_{n}\) must be negative somewhere. Since
\(f''_{n} \neq 0\) for
\(\beta > 0\), if follows that
\(f''_{n} < 0\) for all
\(\beta > 0\). In other words,
\(f_{n}\), defined by (
3.1) is concave from below on
\(0 < \beta < \infty \), i.e., the tangent at any point
\((\beta_{0}, f_{n}(\beta_{0}))\),
\(\beta_{0} > 0\), lies above the graph of
\(f_{n}\) for every
\(\beta > 0\),
\(\beta \neq \beta_{0}\).
Since, by (
3.2),
\(F'_{n} (0) > 0\), if follows that
\(F' > 0\) at least for small positive values of
β. This means that
\(f_{n} - \beta f'_{n} > 0\) for small positive values of
β as can be seen from (
3.3). This can also be verified by means of (
3.6), (
3.7), and (
3.8), respectively. By (
3.6) and (
3.8) we see that
\(f_{n} - \beta f'_{n} > 0\) also for large values of
β, so that
\(F'_{n} > 0\) for large
β. Suppose now that
\(f_{n} - \beta f'_{n} < 0\) at some point
\(\beta > 0\). Then there exist points
\(\beta_{1}\) and
\(\beta_{2}\),
\(0 < \beta_{1} < \beta_{2}\), such that
$$\begin{aligned}& f_{n} (\beta_{\nu }) = \beta_{\nu }f'_{n} (\beta_{ \nu }), \quad \nu = 1, 2, \end{aligned}$$
(3.9)
\(f_{n} - \beta f'_{n} > 0\) for
\(0 < \beta < \beta_{1}\) and
\(\beta_{2} < \beta < \infty \), and
\(f_{n} - \beta f'_{n} < 0\) for
\(\beta_{1} < \beta < \beta_{2}\). Consequently, there would exist a continuous function
\(\alpha (\beta)\) such that
\(\alpha (\beta_{\nu }) = 0\),
\(\nu = 1, 2\),
\(\alpha (\beta) > 0\),
\(\beta_{1} < \beta < \beta_{2}\), and
$$f'_{n} - \frac{1}{\beta } f_{n} - \alpha = 0, \quad \beta_{1} < \beta < \beta _{2}. $$
This inhomogeneous linear differential equation has the unique solution
$$\begin{aligned}& f_{n} = \biggl( \frac{f_{n} (\beta_{1})}{\beta_{1}} \biggr) \beta + \beta \int _{\beta_{1}}^{\beta } \frac{\alpha (\tau)}{\tau } \,d\tau, \quad \beta_{1} \le \beta \le \beta_{2}, \end{aligned}$$
(3.10)
with initial condition
\((\beta_{1}, f_{n} (\beta_{1}))\). Its derivative is
$$f'_{n} = \frac{f_{n} (\beta_{1})}{\beta_{1}} + \int _{\beta_{1}}^{\beta } \frac{\alpha (\tau)}{\tau } \,d\tau + \alpha ( \beta), \quad \beta_{1} \le \beta \le \beta_{2}. $$
Observing (
3.9) for
\(\nu =2\) and noting that
\(\alpha (\beta_{2}) =0\), we have at
\(\beta_{2}\)
$$\begin{aligned}& f'_{n} (\beta_{2}) = \frac{f_{n} (\beta_{1})}{\beta_{1}} + \int _{\beta_{1}}^{\beta _{2}} \frac{\alpha (\tau)}{\tau } \,d\tau = \frac{f_{n} (\beta_{2})}{\beta_{2}} > \frac{f_{n} (\beta_{1})}{\beta_{1}}, \end{aligned}$$
(3.11)
since the integral is positive. Now, the tangent to the integral curve defined by (
3.10) at the point
\((\beta_{1}, f_{n} (\beta_{1}))\) is given by
\(y(\beta) = f_{n} (\beta_{1}) + (\beta - \beta_{1}) f'_{n} (\beta _{1})\) with
\(f'_{n} (\beta_{1}) = f_{n}(\beta_{1}) / \beta_{1}\) by (
3.9).
Thus,
$$y(\beta_{2}) = f_{n} (\beta_{1}) + ( \beta_{2} - \beta_{1}) \frac{f_{n} (\beta_{1})}{\beta_{1}} = \frac{f_{n} (\beta_{1})}{\beta_{1}} \beta_{2}. $$
Since
\(f_{\nu }\) is concave from below it follows that
$$y(\beta_{2}) = \frac{f_{n} (\beta_{1})}{\beta_{1}} \beta_{2} > f_{n} (\beta_{2}), $$
or
$$\frac{f_{n} (\beta_{2})}{\beta_{2}} < \frac{f_{n} (\beta_{1})}{\beta_{1}} $$
in contradiction to (
3.11). Consequently,
\(f_{n} - \beta f'_{n} > 0\) for all
\(\beta \in (0, \infty)\), and, hence, as (
2.3) shows,
\(F'_{n} > 0\) for
\(0 < \beta < \infty \), which means that
\(F_{n} = Q_{n} / S_{n}\), or
\(F_{n} = (xf_{n} + \beta) / [ (x+2)f_{n} + \beta ]^{-1}\), is a strictly monotonically increasing function of
\(\beta \in (0, \infty)\). A corresponding result holds for
\(G_{n} = T_{n} / S_{n}\).
We now show that the sequence
\(\{ F'_{n} \}\) converges uniformly on every closed interval
\([a, b] \subset (0, \infty)\). Differentiating
\(F_{n} = Q_{n} / S_{n}\), we form
$$\begin{aligned}& \bigl\vert F'_{n+k} - F'_{n} \bigr\vert = \bigl(S_{n}^{2} S_{n+k}^{2}\bigr)^{-1} \bigl\vert S _{n}^{2} \bigl(Q'_{n+k} S_{n+k} - Q_{n+k} S'_{n+k}\bigr) - S_{n+k}^{2} \bigl(Q'_{n} S _{n} - Q_{n} S'_{n}\bigr) \bigr\vert . \end{aligned}$$
(3.12)
With
\(Q = \sum q_{\nu }\),
\(S = \sum s_{\nu }\), equation (
3.12), after some manipulations, can be brought into the form
$$\begin{aligned}& \textstyle\begin{cases} \vert F'_{n+k} - F'_{n} \vert \\ \quad = ( S_{n}^{2} S_{n+k}^{2} ) ^{-1} \vert ( \sum_{\nu = n+1}^{n+k} s_{\nu } ) ( -S _{n}^{2} Q'_{n} + 2Q_{n} S_{n} S'_{n} + S_{n}^{2} ( \sum_{\nu = n+1}^{n+k} q'_{\nu } ) ) \\ \qquad {} + ( \sum_{\nu = n+1}^{n+k} s_{\nu } ) ^{2} ( Q'_{n} S_{n} - Q_{n} S'_{n} ) \\ \qquad {} - ( \sum_{\nu = n+1}^{n+k} s'_{ \nu } ) ( S_{n}^{2} Q_{n} + S_{n}^{2} ( \sum_{\nu = n+1}^{n+k} q_{\nu } ) ) \\ \qquad {} - ( \sum_{\nu = n+1}^{n+k} q_{\nu } ) S_{n} ^{2} S'_{n} + ( \sum_{\nu = n+1}^{n+k} q'_{\nu } ) S _{n}^{3} \vert . \end{cases}\displaystyle \end{aligned}$$
(3.13)
Here, we replace all negative terms between the absolute value bars by their absolute values. Since
Q,
\(|Q'|\),
S,
\(|S'|\) are series with positive terms, we increase the right-hand side of (
3.13) by replacing their partial sums by the entire series. Then we remember the fact that these series are monotonically decreasing functions of
β. Therefore, we increase the right-hand side of (
3.13) further by setting
\(Q_{n} < Q < Q(a)\),
\(|Q'_{n}| < |Q'| < |Q'(a)|\),
\(S_{n} < S(a)\),
\(|S'_{n}| < |S'(a)|\) for
\(\beta \in [a,b]\). We may also replace
\(\sum q_{\nu }\) by
\(\sum s_{\nu }\) since
\(0 < q_{\nu }< s_{\nu }\). Thus, there exist positive constants
\(K_{\nu }\) (
\(\nu = 1,\dots,5\)) such that equality (
3.13) may be replaced by the inequality
$$\begin{aligned}& \textstyle\begin{cases} \vert F'_{n+k} - F'_{n} \vert < ( S_{n}^{2} S_{n+k}^{2} ) ^{-1} \{ K_{1} \sum_{\nu = n+1}^{n+k} s_{\nu }+ K_{2} ( \sum_{\nu = n+1}^{n+k} s_{\nu } ) ^{2} + K_{3} \sum_{\nu = n+1}^{n+k} \vert s'_{\nu } \vert \\ \hphantom{ \vert F'_{n+k} - F'_{n} \vert < } {} + K_{4} \sum_{\nu = n+1}^{n+k} s_{\nu }+ K_{5} \sum_{\nu = n+1}^{n+k} \vert q'_{\nu } \vert \} . \end{cases}\displaystyle \end{aligned}$$
(3.14)
Furthermore, by (
1.11),
$$\begin{aligned} S_{n+k} >{} & S_{n} > S_{1} = \bigl[ (\beta + x -1) (\beta + x) (\beta + x +1) \bigr] ^{-1} \\ >{} & \bigl[ (b+x-1) (b+x) (b+x+1) \bigr] ^{-1} = B^{-1} > 0, \quad \beta \in [a,b], \end{aligned}$$
so that
$$\begin{aligned}& \bigl( S_{n}^{2} S_{n+k}^{2} \bigr) ^{-1} < \bigl( S _{n}^{4} \bigr) ^{-1} < \bigl( S_{1}^{4} \bigr) ^{-1} < B^{4},\quad \beta \in [a, b]. \end{aligned}$$
(3.15)
Now, since
S,
\(S'\), and
\(Q'\) converge uniformly on
\([a, b] \in (0, \infty)\), given
\(\varepsilon > 0\) there exists a number
\(n_{0}( \varepsilon)\) such that each of the five finite sums in (
3.14) is less than
\(B^{-4} (\sum_{\nu =1}^{5} K_{\nu })^{-1} \varepsilon \) for every
\(n \geq n_{0}\) and for all
\(k \geq 1\). Thus (
3.14) together with (
3.15) leads to
\(|F'_{n+k} - F'_{n}| < \varepsilon \) for every
\(n \geq n_{0}\) and for all
\(k \geq 1\) on every
\([a, b] \subset (0, \infty)\). The final result is that
\(\{ F'_{n} \}\) converges uniformly for
\(\beta \in [a,b]\) to the function
\((Q/S)' > 0\), and this means that
\(Q/S\) is strictly monotonically increasing. (This result justified the direction of the arrows in (
2.14) and (
2.20), and in (
2.21) and (
2.22) for
\(T/S\).)