Sharp Inequalities for Trigonometric Functions

and Applied Analysis 3 Differentiation and using (14) and (15) yield

The main purpose of this paper is to find the sharp bounds for the functions   cot −1 ( ∈ (0, /2)), which include the corresponding trigonometric version of the inequalities listed above.As applications, their corresponding inequalities for bivariate means are presented.
Making use of the monotonicity of   and the facts that we get inequality (27) and its reverse immediately.
Similarly, we get which proves the desired result.

Main Results
3.1.The First Sharp Bounds for   cot −1 .In this subsection, we present the sharp bounds for   cot −1 in terms of (cos ) 1/ , which give the trigonometric versions of inequalities ( 6) and (7).
The above inequalities can be rewritten as where the equality is due to the fact that  1 is the unique root of (42).Therefore, we get the right inequality in (41) and the first inequality in (44).We clearly see that  1 is the best possible constant.

The Second Sharp
Bounds for   cot −1 .In this subsection, we give the sharp bounds for   cot −1 in terms of (cos ) 2/(3 2 ) , which give the trigonometric versions of inequalities (8).
It remains to prove that  2 is the best possible constant.If there is a   2 ∈ (0, 1) with   2 >  2 such that the right inequality in (55) holds for  ∈ (0, /2), then, by the second assertion proved previously, we have    2 (/2 − ) > 0, which yields a contradiction.
(iii) The first and second inequalities in (57) and their reverse ones are clearly the direct consequences of Lemma 5.
From the proof of Theorem 10, we clearly see that the constant 1/ √ 10 in (79) is the best possible constant, but  3 =  2 /2 is not.
Using certain known inequalities and the corollary above, we can obtain the following novel inequalities chain for trigonometric functions.
The fourth one in (82) is equivalent to which holds due to for  ∈ (0, /2).
The ninth one easily follows from The tenth, eleventh, and twelfth ones can be obtained by [19, ( 3.9)].
Except the last one, other ones are obviously deduced from Corollary 12.
The last one is equivalent to which follows from the inequality connecting the fourth and sixth members in (82) proved previously.Thus, the proof is complete.
In fact, Lemma 7 implies that   →   (/2 − ) is increasing on (0, 1), which in conjunction with the facts that indicates the second one.By using mathematical software, we find  2 ≈ 0.5763247.
On the other hand, if there is a  * ∈ (0, 1) with  * >  4 such that the second inequality in (55) holds for  ∈ (0, /2), then by the second assertion proved previously, we have   * (/2 − ) > 0, which leads to a contradiction.This proves that the constant  4 is the best possible constant.
(iii) The first and second inequalities in (57) and their reverse ones are clearly the direct consequences of Lemma 6.It remains to prove the third one.We have to determine the sign of   () defined by for  ∈ (0, /2) and  ∈ (0, 1).Simplifying leads to As shown previously,   (/2 − ) < 0 for  ∈ (0, 4 ) and   (/2 − ) > 0 for  ∈ ( 4 , 1), which in combination with ln(cos ) > ln(cos(/2)) and ln(cos(/2)) < 0 gives the desired result.Lemma 7 reveals the monotonicity of the first, second, and third members in (104) with respect to  on (0, 1) due to Finally, we prove that   is the best possible constant.It can be deduced from lim Thus, the proof is complete.
We note that (102) can be written as Making use of the monotonicity of the function   → (cos ) 1/(3 2 ) on (0, 1) given in Lemma 7 together with Corollary 12 and Theorem 19, we obtain the following.
We now deduce some inequalities involving these means from the inequalities for trigonometric functions established in Section 3.
The following follows from (88).