Research Article On New Wilker-Type Inequalities

Transcription

1 International Scholarly Research Network ISRN Mathematical Analysis Volume 2011, Article ID , 7 pages doi: /2011/ Research Article On New Wilker-Type Inequalities Zhengjie Sun and Ling Zhu Department of Mathematics, Zhejiang Gongshang University, Hangzhou , China Correspondence should be addressed to Ling Zhu, zhuling0571@163.com Received 23 February 2011; Accepted 27 April 2011 Academic Editor: G. Lu Copyright q 2011 Z. Sun and L. Zhu. 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. In this paper, we present and prove a new Wilker-type inequality for hyperbolic functions. We also give a simple proof of the counterpart of the above inequality for the circular functions. 1. Introduction Wilker 1 proposed two open questions as the following statements. Problem 1. If 0 <<π/2, then sin 2 tan > Problem 2. There eists a largest constant c such that sin 2 tan > 2 c 3 tan 1.2 for 0 <<π/2. Sumner et al. 2 affirmed the truth of the problems above and obtained a further result as follows.

2 2 ISRN Mathematical Analysis Theorem 1.1. If 0 <<π/2, then 16 π 4 3 tan < sin 2 tan 2 < tan. 1.3 Furthermore, 16/π 4 and 8/45 are the best constants in 1.3. Guo et al. 3 gave new proofs of inequalities 1.1 and 1.2. In recent years, Zhu 4 showed a new simple proof of inequality 1.1; Pinelis 5 got other proof of inequalities 1.3 by using L Hospital rules for monotonicity; Zhang and Zhu 6 gave a new elementary proof of double inequalities 1.3. Besides, in article 7,Wangoffered another type of the inequality and proved it by the power series epansions of circular functions as follows Theorem 1.2. Let 0 <<π/2. Then, sin < 2 sin tan 2 < π 16 3 sin. π Furthermore, 2/45 and 2/π 16/π 3 are the best constants in 1.4. Zhu 8 established a new Wilker-type inequality involving hyperbolic functions as follows: sinh 2 tanh 2 > tanh, > 0, 1.5 where the constant 8/45 is the optimum constant in 1.5. In fact, we can show a new Wilker-type inequality involving hyperbolic functions referring to the result above. Theorem 1.3. Let >0. Then 2 sinh tanh 2 < sinh. 1.6 Furthermore, 2/45 is the best constant in 1.6. The purpose of this paper is to present a concise proof of inequality 1.6 by using the power series epansion of hyperbolic functions and to give an elementary proof of inequality 1.4 in another way. 2. The Proof of Theorem 1.3 Let / sinh 2 / tanh 2/ 3 sinh 2 sinh cosh 2sinh 2 / sinh 3 A/B, where A 2 sinh cosh 2sinh 2,andB sinh 3.Wegetthat

3 ISRN Mathematical Analysis 3 the eistence of Theorem 1.3 can be ensured when proving the following two statements: the first is A/B 2/45 or 2B 45A, and the second is lim 0 A/B 2/45. Since A sinh cosh n1 2n 1! 2 2n 2n! n0 2 2n1 2n 1! 2n2 n0 2 2n 2n! 1 n0 n0 1 n1 n1 2 2n 2n 1! 2n2 n1 2 2n2 2n 2! 2n2 2 2n 2n 1! 22n2 2n2 2n 2! a n 2n2, n2 2 2n 2n 1! n2 22n2 2n2 2n 2! 2.1 B sinh 3 3sinh 1 3 2n n 1! 2n n n 1! 2n4 n1 n n n n 1! 2n4 42n 1! 2n2 b n 2n2, n2 n2 n0 where a n 2 2n /2n 1! 2 2n2 /2n 2!, b n 3 2n 1 3/42n 1!, n 2, and n N, according to the analysis above, the proof of Theorem 1.3 can be completed when proving that 2b n 45a n, 2.2 that is, nn 12n 19 n 9 135n 14 n 2.3 for n 2.

4 4 ISRN Mathematical Analysis Let f for 2,. We compute f log log 4, f log log log log 4 2, f log log log log log 4 3, f 4 a 3 b 2 c d 9 e m4, 2.4 where a 2 log 9 4 > 0, b 3 log log 9 3 > 0, c log log log 9 2 > 0, d 4 log log log 9 > 0, 2.5 e 135 log 4 4 > 0, m 540 log log > 0. Then let H a 3 b 2 c d9/4 /e m; we compute H l/e m 2 9/4, where l a 4 4 a 3 3 a 2 2 a 1 a 0, a 4 ae log 9 4 > 0, a 3 2ae eb am log 9 4 > 0, a 2 be 3am ec mb log 9 4 > 0, 2.6 a 1 2bm mc de log 9 4 > 0, a 0 m c d log 9 ed > 0. 4

5 ISRN Mathematical Analysis 5 As we can see, all coefficients of the polynomial l are positive integers. When >2, we have l > 0andH > 0. In view of that H > 1,f > 0,f > 0,f > 0,f > 0, we have that H > 1andf 4 > 0for >2. We can also conclude that f i i 3, 2, 1, 0 is increasing on 2,, respectively. Since f2 0, we have f 0for 2. So the proof of inequality 2.3 is completed. Furthermore, lim 0 A/B a 2 /b /5! 2 6 /6!/3 3 3/4 3! 2/45; the proof of Theorem 1.3 is completed. 3. Another Proof of Theorem 1.2 We simplify the double inequality 1.4 into another form: 2 45 < / sin 2 / tan 2 < 2 3 sin π 16 π, 0 <<π Here we will discuss the monotonicity of the function g / sin 2 / tan 2/ 3 sin 1/ sin / sin 2 cos /sin 2 2/ sin / 3. If the function g is increasing for 0 <<π/2 and both lim 0 g 2/45 and lim π/2 g 2/π 16/π 3 are right, the proof of equality 1.4 is completed. From 9 or 10, page 75, we know that for 0 < <πthe equality 1 sin 1 2 2n 2 2n! B 2n 2n 1 n1 3.2 holds. So we can also know that for 0 < <π cos sin sin 1 sin 3 cos 2sin 2 2 n1 1 2sin n2 2 2n 2 2n! B 2n 2n 1 2n 2, 2 2n 2 2n! B 2n 2n 12n 2 2n n 2 2 2n! B 2n 2n 1 n1

6 6 ISRN Mathematical Analysis hold. Then we compute g n2 a n 2n 4 by using the results above, where a n 1 2 2n 2 2 2n! B 2n 2n 12n n 2 2 2n 2! B 2n 2 22n 2 2n! B 2n 2n n 2 2n! B 2n 1 2 2n n 2! B 2n n n 2! B 2n 2 2n 12n 2 22n n 2 B 2n 22n! 2n2n 5 2 2n 2 B 2n. 22n! 3.4 When n 2, we can compute a 2 1/22/2! B 2 4/2 4! 14 B 4 1/2 1/6 7/6 1/30 2/45 > 0byusing B 2 1/6 and B 4 1/30. Furthermore, there is an obvious fact that a n > 0forn 3. So far, we can demonstrate that a n > 0 for all n 2. So the function g is increasing on 0,π/2. Evidently, lim 0 g 2/45 and lim π/2 g 2/π 16/π 3 are true. So the proof of the double inequality of 1.4 is completed. Remark 3.1. Wilker s inequalities 1.1 and 1.2 have been further refined by many scholars in the past few years; the readers can refer to References 1 J. B. Wilker, E3306, The American Mathematical Monthly, vol. 96, p. 55, J. S. Sumner, A. A. Jagers, M. Vowe, and J. Anglesio, Inequalities involving trigonometric functions, The American Mathematical Monthly, vol. 98, no. 3, pp , B.-N. Guo, B.-M. Qiao, F. Qi, and W. Li, On new proofs of Wilker s inequalities involving trigonometric functions, Mathematical Inequalities and Applications, vol. 6, no. 1, pp , L. Zhu, A new simple proof of Wilker s inequality, Mathematical Inequalities and Applications, vol. 8, no. 4, pp , I. Pinelis, l Hospital rules for monotonicity and the Wilker-Anglesio inequality, The American Mathematical Monthly, vol. 111, no. 10, pp , L. Zhang and L. Zhu, A new elementary proof of Wilker s inequalities, Mathematical Inequalities and Applications, vol. 11, no. 1, pp , Z.-H. Wang, A new Wilker-type inequality, Yibing University, vol. 6, pp , L. Zhu, On Wilker-type inequalities, Mathematical Inequalities and Applications, vol. 10, no. 4, pp , J.-L. Li, An identity related to Jordan s inequality, International Mathematics and Mathematical Sciences, Article ID 76782, M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, U.S. National Bureau of Standards, Washington, DC, USA, L. Zhu, Some new Wilker-type inequalities for circular and hyperbolic functions, Abstract and Applied Analysis, vol. 2009, Article ID , S.-H. Wu and H. M. Srivastava, A further refinement of Wilker s inequality, Integral Transforms and Special Functions, vol. 19, no. 9-10, pp , E. Neuman, One- and two-sided inequalities for Jacobian elliptic functions and related results, Integral Transforms and Special Functions, vol. 21, no. 5-6, pp , E. Neuman, Inequalities involving inverse circular and inverse hyperbolic functions II, Mathematical Inequalities, vol. 4, no. 1, pp , 2010.

7 ISRN Mathematical Analysis 7 15 E. Neuman and J. Sandor, On some inequalities involving trigonometric and hyperbolic functions with emphasis on the Cusa-Huygens, Wilker, and Huygens inequalities, Mathematical Inequalities and Applications, vol. 13, no. 4, pp , S.-H. Wu and H. M. Srivastava, A weighted and eponential generalization of Wilker s inequality and its applications, Integral Transforms and Special Functions, vol. 18, no. 7-8, pp , 2007.

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