Monotonicity rule for the quotient of two functions and its application

In the article, we provide a monotonicity rule for the function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$[P(x)+A(x)]/[P(x)+B(x)]$\end{document}[P(x)+A(x)]/[P(x)+B(x)], where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$P(x)$\end{document}P(x) is a positive differentiable and decreasing function defined on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(-R, R)$\end{document}(−R,R) (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$R>0$\end{document}R>0), and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A(x)=\sum ^{\infty}_{n=n_{0}}a_{n}x^{n}$\end{document}A(x)=∑n=n0∞anxn and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B(x)=\sum^{\infty }_{n=n_{0}}b_{n}x^{n}$\end{document}B(x)=∑n=n0∞bnxn are two real power series converging on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(-R, R)$\end{document}(−R,R) such that the sequence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\{a_{n}/b_{n}\}_{n=n_{0}}^{\infty}$\end{document}{an/bn}n=n0∞ is increasing (decreasing) with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a_{n_{0}}/b_{n_{0}}\geq(\leq)\ 1$\end{document}an0/bn0≥(≤)1 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$b_{n}>0$\end{document}bn>0 for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n\geq n_{0}$\end{document}n≥n0. As applications, we present new bounds for the complete elliptic integral \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{E}(r)=\int_{0}^{\pi /2}\sqrt{1-r^{2}\sin^{2}t}\,dt$\end{document}E(r)=∫0π/21−r2sin2tdt (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0< r<1$\end{document}0


Introduction
The most commonly used monotonicity rule in elementary calculus is that f is increasing (decreasing) on [a, b] if f : [a, b] → R is continuous on [a, b] and has a positive (negative) derivative on (a, b), and it can be proved easily by the Lagrange mean value theorem. The functions whose monotonicity we prove in this way are usually polynomials, rational functions, or other elementary functions. But we often find that the derivative of a quotient of two functions is quite messy and the process is tedious. Therefore, the improvements, generalizations and refinements of the method for proving monotonicity of quotients have attracted the attention of many researchers.
In , Biernacki and Krzyż [] (see also [], Lemma ., []) found an important criterion for the monotonicity of the quotient of power series as follows.

Theorem . ([]) Let A(t) = ∞
k= a k t k and B(t) = ∞ k= b k t k be two real power series converging on (-r, r) (r > ) with b k >  for all k. If the non-constant sequence {a k /b k } is increasing (decreasing) for all k, then the function t → A(t)/B(t) is strictly increasing (decreasing) on (, r).  [], Theorem .) established l'Hôpital's monotone rule that can be applied to a wide class of quotients of functions. , .
is strictly monotone, then the monotonicity in the conclusion is also strict.

Theorem . ([]) Let f and g be differentiable and g never vanish on an open interval
Then the following statements are true: Recently, Yang et al.
[], Theorem ., established a more general monotonicity rule for the ratio of two power series.
k= a k t k and B(t) = ∞ k= b k t k be two real power series converging on (-r, r) and b k >  for all k, and H A,B = A B/B -A. Suppose that for certain m ∈ N, the non-constant sequence {a k /b k } is increasing (decreasing) for  ≤ k ≤ m and decreasing (increasing) for k ≥ m. Then the function A/B is strictly increasing (decreasing) on (, r) if and only if H A,B (r -) ≥ (≤) . Moreover, if H A,B (r -) < (>), then there exists t  ∈ (, r) such that the function A/B is strictly increasing (decreasing) on (, t  ) and strictly decreasing (increasing) on (t  , r).
The foregoing monotonicity rules have been used very effectively in the study of special functions [-], differential geometry [, ], probability [] and approximation theory []. The main purpose of the article is to present the monotonicity rule for the function [P(x) + ∞ n=n  a n x n ]/[P(x) + ∞ n=n  b n x n ] and to provide new bounds for the complete elliptic integral of the second kind. Some complicated computations are carried out using Mathematica computer algebra system.

Monotonicity rule
Theorem . Let P(x) be a positive differentiable and decreasing function defined on (, r) (r > ), let A(x) = ∞ n=n  a n x n and B(x) = ∞ n=n  b n x n be two real power series converging on (-r, r). If a n  /b n  ≥ (≤) , b n >  for all n ≥ n  and the non-constant sequence Proof Let x ∈ (, r), and gives Note that I  can be rewritten as If a n  /b n  ≥ (≤) , b n >  for all n ≥ n  and the non-constant sequence {a n /b n } ∞ n=n  is increasing (decreasing), then we clearly see that for all x ∈ (, r).
It follows from P(x) is a positive differentiable and decreasing function on (, r) that follows easily from (.)-(.), and the proof of Theorem . is completed.

Bounds for the complete elliptic integral of the second kind
For r ∈ (, ), Legendre's complete elliptic integral [] of the second kind is given by It is well known that E( + ) = π/, E( -) = , and E(r) is the particular case of the Gaussian hypergeometric function Recently, the bounds for the complete elliptic integral E(r) of the second kind have been the subject of intensive research. In particular, many remarkable inequalities for E(r) can be found in the literature [-]. Vuorinen [] conjectured that the inequality holds for all r ∈ (, ), where, and in what follows, r = (-r  ) / . Inequality (.) was proved by Barnard In this section, we shall use Theorem . to present new bounds for the complete elliptic integral E(r) of the second kind. In order to prove our main result, we need three lemmas, which we present in this section. Lemma . (see [], Lemma ) Let n ∈ N and m ∈ N ∪ {} with n > m, a i ≥  for all  ≤ i ≤ n, a n a m >  and Then there exists t  ∈ (, ∞) such that P n (t  ) = , P n (t) <  for t ∈ (, t  ) and P n (t) >  for t ∈ (t  , ∞).
respectively. Then w n ≥  for all n ≥ .
Proof Let p  (n), p  (n), p  (n), α n and β n be defined by respectively. Then from (.)-(.) and elaborated computations we get n ≥ , we clearly see that there exists n  >  such that the sequence {α n /β n } ∞ n= is increasing for  ≤ n ≤ n  and decreasing for n ≥ n  , which implies that It follows from Lemma . that for all n ≥ . Therefore, Lemma . follows easily from (.)-(.) and (.) together with the facts that p  (n) > , p  (n + ) >  and p  (n + ) >  for n ≥ .

Remark . Let
where J(r ) is defined by (.). Then simple computations lead to From (.), (.), (.) and (.)-(.) we clearly see that there exists small enough δ ∈ (, ) such that the lower bound given in (.) for E(r) is better than the lower bound given in (.) for r ∈ (δ, δ), the lower bound given in (.) for E(r) is better than the lower bound given in (.) for r ∈ (, δ), the upper bound given in (.) for E(r) is better than the upper bound given in (.) for r ∈ (, δ), and the upper bound given in (.) for E(r) is better than the upper bound given in (.) for r ∈ (δ, δ).

Corollary . Let J(r ) be defined by (.). Then the double inequality
holds for all r ∈ (, ).
Proof Let F(r) be defined by (.) and Then we clearly see that From (.) and the proof of Theorem . we know that F(r) is strictly decreasing on (, ) and A(r) is strictly increasing on (, ). Therefore, inequality (.) follows from (.) and (.) together with the monotonicity of A(r) on the interval (, ).

Corollary . Let J(r ) be defined by (.). Then the double inequality
holds for all r ∈ (, ). for all r ∈ (, ), which implies that both the absolute and relative errors using πJ(r )/ to approximate E(r) are less than .%.

Conclusions
In this paper, we find a monotonicity rule for the function [P(x) + ∞ n=n  a n x n ]/[P(x) + ∞ n=n  b n x n ]. As applications, we present new bounds for the complete elliptic integral E(r) = π /  √ r  sin  t dt ( < r < ) of the second kind, and we show that our bounds are sharper than the previously known bounds for some r ∈ (, ).