Sharp Inequalities for Trigonometric Functions

Abstract and Applied Analysis, Jul 2014

We establish several sharp inequalities for trigonometric functions and present their corresponding inequalities for bivariate means.

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Sharp Inequalities for Trigonometric Functions

Sharp Inequalities for Trigonometric Functions Zhen-Hang Yang,1 Yun-Liang Jiang,2 Ying-Qing Song,1 and Yu-Ming Chu1 1School of Mathematics and Computation Sciences, Hunan City University, Yiyang 413000, China 2School of Information Engineering, Huzhou Teachers College, Huzhou 313000, China Received 4 March 2014; Revised 21 May 2014; Accepted 30 May 2014; Published 7 July 2014 Academic Editor: Chun-Gang Zhu Copyright © 2014 Zhen-Hang Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We establish several sharp inequalities for trigonometric functions and present their corresponding inequalities for bivariate means. 1. Introduction A bivariate real value function is said to be a mean if for all . is said to be homogeneous if for any . Remark 1 (see [1]). Let be a homogeneous bivariate mean of two positive real numbers and . Then where . By this remark, almost all of the inequalities for homogeneous symmetric bivariate means can be transformed equivalently into the corresponding inequalities for hyperbolic functions and vice versa. More specifically, let , , and be the logarithmic, identric, and th power means of two distinct positive real numbers and given by respectively. Then, for , we have where . By Remark 1, we can derive some inequalities for hyperbolic functions from certain known inequalities for bivariate means mentioned previously. For example, (see [2, 3]); consider (see [4, 5]); consider that (see [1]) holds for if and only if and ; consider (see [6]); consider (see [7], ( 3.9), and ( 3.10)); if , then the double inequality (see [8]) holds if and only if and ; if , then inequality (11) holds if and only if and ; consider that (see [9]) holds if and only if and . The main purpose of this paper is to find the sharp bounds for the functions , which include the corresponding trigonometric version of the inequalities listed above. As applications, their corresponding inequalities for bivariate means are presented. 2. Lemmas Lemma 2 (see [10, Theorem 1.25], [11, Remark 1]). For , let be continuous on and differentiable on ; let on . If is increasing (or  decreasing) on , then so are If is one-to-one, then the monotonicity in the conclusion is strict. Lemma 3 (see [12]). Let and    be real numbers and let the power series and be convergent for . If , for , and is increasing (decreasing), for , then the function is also (strictly) increasing (decreasing) on . Lemma 4 (see [13, pages 227–229]). One has where is the Bernoulli number. Lemma 5. For every , , the function defined by is increasing if and decreasing if . Consequently, for , one has It is reversed if . Proof. For , we define and , where . Note that , and can be written as Differentiation and using (14) and (15) yield where Clearly, if the monotonicity of is proved, then by Lemma 3 we can get the monotonicity of , and then the monotonicity of the function easily follows from Lemma 2. For this purpose, since , , for , we only need to show that is decreasing if and increasing if . Indeed, an elementary computation yields It is easy to obtain that, for , which proves the monotonicity of . Making use of the monotonicity of and the facts that we get inequality (19) and its reverse immediately. Lemma 6. For every , , the function defined by is increasing if and decreasing if . Consequently, for , one has It is reversed if . Proof. We define and , where . Note that , and can be written as Differentiating and using (14) and (15) yield where Similarly, we only need to show that is decreasing if and increasing if . In fact, simple computation leads to It is easy to obtain that, for , which proves the monotonicity of . Making use of the monotonicity of and the facts that we get inequality (27) and its reverse immediately. Lemma 7 (see [14, 15]). For and , let , , , and be defined by Then, , , and are decreasing with respect to , while is increasing with respect to on . Proof. It was proved in [14, 15] that the functions and are decreasing with respect to . Now, we prove that has the same property. Logarithmic differentiation gives that, for , Clearly, for and , which yields , and so . This gives and . Similarly, we get which implies that is decreasing with respect to on . Therefore, which proves the desired result. 3. Main Results3.1. The First Sharp Bounds for In this subsection, we present the sharp bounds for in terms of , which give the trigonometric versions of inequalities (6) and (7). Theorem 8. For , the two-side inequality holds with the best possible constants and , where is the unique root of the equation on . Moreover, one has where the exponents , and coefficients , in (43) are the best possible constants and so is in (44). Proof. (i) We first prov (...truncated)


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Zhen-Hang Yang, Yun-Liang Jiang, Ying-Qing Song, Yu-Ming Chu. Sharp Inequalities for Trigonometric Functions, Abstract and Applied Analysis, 2014, 2014, DOI: 10.1155/2014/601839