Sharp inequalities for tangent function with applications

In the article, we present new bounds for the function etcot(t)−1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$e^{t\cot(t)-1}$\end{document} on the interval (0,π/2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(0, \pi/2)$\end{document} and find sharp estimations for the Sine integral and the Catalan constant based on a new monotonicity criterion for the quotient of power series, which refine the Redheffer and Becker-Stark type inequalities for tangent function.


Introduction
The study of this paper is concerned with the following inequality: which was posted by Redheffer in [] and was proved by Williams []. Recently, Zhu and Sun [] extended the Redheffer inequality (.) to the tangent function, and they established the following inequalities: π  + t  π  -t  π  / < tan t t < π  + t  π  -t  , t ∈ (, π/) (.) with the best exponents π  / and . Zhu [] further refined the double inequality It is worth noting that Becker and Stark [] in  showed the double inequality  π  -t  < tan t t < π  π  -t  , t ∈ (, π/), (.) where  and π  are the best constants. Later, Zhu and Hua [] gave a general refinement of the Becker-Stark inequalities (.) by the power series expansion of the tangent function in terms of the Bernoulli numbers. In particular, they proved that for t ∈ (, π/) the double inequality π  + (/π  -)t  π  -t  < tan t t < π  + (π  / -)t  π  -t  (.) holds with the best constants (/π  -) and (π  / -); also see []. Chen and Cheung [] further presented an improvement of the left hand side inequality in (.), which states that holds for t ∈ (, π/) with the best exponents α = π  / and β =  (also cf. []). Another improvement involving the left hand side one in (.) was made in [] by Nishizawa. Very recently, Bhayo and Sándor [], Corollary , again proved the Becker-Stark inequalities (.) by using Redheffer inequality (.), which reveals the implicit relation between Redheffer's and Becker-Stark's inequalities. They in [], Corollaries , also stated that for t ∈ (, π/) we have It is an important observation that Yang et al. [], (), in  considered the bounds for function e t cot t- and established a number of inequalities for trigonometric functions.
In particular, they in [], Corollary , showed that for t ∈ (, π/) e -t  /π  < e t cot t- < e -t  / , which can be written as Inspired by these results mentioned above, the aim of this paper is to determine the best bounds for Y (t) = e t cot t- in terms of on (, π/), that is to say, we will determine the best parameters p, q ∈ (-∞, /π  ] such that the double inequality holds for all t ∈ (, π/). Inequalities (.) also can be rewritten as which offers a new type of bounds being different of the previous papers for the tangent function.

Some useful lemmas
In order to prove the main Theorem  in the next section, we need some preliminary lemmas. To this end, we first introduce a useful auxiliary function H f ,g . For -∞ ≤ a < b ≤ ∞, let f and g be differentiable on (a, b) and g =  on (a, b). Then the function H f ,g is defined by The function H f ,g has been investigated with some well properties in [], Properties , , which plays an important role in the proof of a monotonicity criterion for the quotient of power series; also see [].
where B n is the Bernoulli number.
Proof To avoid complicated calculations, we here make use of Lemmas ,  and  to prove this lemma. For this purpose, we write g(t) as then applying Lemma  yields We now prove that the sequence {a n /b n } is increasing for  ≤ n ≤  and decreasing for n ≥ . A simple check yields and it remains to show that a n- /b n- > a n /b n for n ≥ . Indeed, we have a n- b n- a n b n =  (n -)(n -) Then by Lemma , we get a n- b n- a n b n - > n(n + ) (n -)(n -) n(n -) (n + )(n + ) where the inequality holds due to  n+ -n  >  for n ≥ . This proves the piecewise monotonicity of {a n /b n } n≥ . According to Lemma , we also have to check that H g  ,g  (π -) <  and H g  ,g  (π/) = . In fact, we have then Lemma  leads to the result that there is a unique t  ∈ (, π) such that g is increasing on (, t  ) and decreasing on (t  , π). Note that we clearly see that the unique t  = π/. A simple computation yields which completes the proof.
Remark  If we use an ordinary method to prove the piecewise monotonicity of g, then it is very troublesome. For example, a direct computation yields As a result, there are various approaches to showing the piecewise monotonicity of g  on (, π), but it seems to be difficult. It thus can be seen that our method used previously is relatively easy.

t) are strictly decreasing and increasing on
where Differentiation again leads to are strictly decreasing and increasing on (-∞, /π  ], respectively.

Main results
This section is devoted to stating and proving the main results concerning some inequalities for the tangent function. More precisely, we have the following.
Proof Let Differentiation yields where g(t) is defined by (.).

Corollary  For t ∈ (, π/), the inequalities
hold with the best coefficients
Proof Clearly, the sufficiency easily follows by Theorem . The necessary condition for the right hand side inequality in (.) to hold for t ∈ (, π/) follows from the limit relation The necessary condition for the left hand side inequality in (.) to hold for t ∈ (, π/) can be obtained from the inequality It follows from Lemma  that p ≥ p  , which completes the proof.

Comparisons and remarks
By Theorem , we have where p  ≈ ..
(ii) The second inequality directly follows from The third one is deduced by for t ∈ (, π/).
(iv) Finally, we prove that ZS(t), Z(t) and BS  (t) are not comparable with each other for all t ∈ (, π/). Simple computations yield This completes the proof. Remark  We claim that the result stated in Theorem  is stronger than the inequality (.), that is, for t ∈ (, π/), we have the inequalities

Remark 
Indeed, the right hand side for this inequality in (.) follows from Corollary , while the left hand side one is the inequality connecting the first and third bounds in (.).
Remark  Lemma  tells us that   < t - cos t sin t -t sin  t t  (t -sin t cos t) <  π  for t ∈ (, π/), Then from equation (.) we find that for f (t) <  for t ∈ (, π) when p = /π  , and so f (t) < f ( + ) = . This gives the following inequality: for all t ∈ (, π), which can be stated as the following proposition.
Proposition  For all t ∈ (, π), we have Remark  The inequality is true due to Cusa and Huygens' paper (see, e.g. []), which is now known as Cusa's inequality (see e.g. [, -]). Some refinements and generalizations of Cusa's inequality can be found in [, , , -]. Now by letting t = x/ and simplifying, inequalities (.) and (.) can be written as which give stronger versions of Cusa's inequality.

Proposition  We have
Then by inequalities (.) for p = /(π  ) and q = /, we obtain for t ∈ (, π/). Further, the right hand side inequalities can be improved as follows.

Proposition  The inequalities
hold for t ∈ (, π/) with the best constants s = / √ , r = / and where s = / √ . Differentiation yields Expanding in power series leads to This indicates that h(π/) > h(t) > h( + ) =  for t ∈ (, π/), which proves the second and third inequalities of (.). Considering the limit it is seen that s = / √  and λ s are the best possible constants. The first and fourth ones are derived from the decreasing property of f (st) ≡ f (u) for u ∈ (, sπ/) ⊂ (, π/) proved in Theorem  for p = r/s  = /, and then r = / and ρ r are also the best. This completes the proof.

Applications
In this section, we give some precise estimations for the Sine integral and Catalan constant. The Sine integral is defined by There are many interesting results concerning the Sine integral; see [, -] and the references therein. Now we shall give more accurate estimations.
Note that We are now in the position to evaluate the integral x  ln(sin t) dt for x ∈ (, π/).