Lazarević and Cusa type inequalities for hyperbolic functions with two parameters and their applications

Transcription

1 Yang and Chu Journal of Inequalities and Applications :403 DOI /s R E S E A R C H Open Access Lazarević and Cusa type inequalities for hyperbolic functions with two parameters and their applications Zhen-Hang Yang 1* and Yu-Ming Chu 2 * Correspondence: yzhkm@163.com 1 School of Mathematics and Computation Sciences, Hunan City University, Yiyang, , China Full list of author information is available at the end of the article Abstract In the article, we establish several Lazarević and Cusa type inequalities involving the hyperbolic sine and cosine functions with two parameters. As applications, we find some new bounds for certain bivariate means. MSC: 26D05; 26D07; 33B10; 26E60 Keywords: Lazarević inequality; Cusa type inequality; hyperbolic function; Schwab-Borchardt mean 1 Introduction The well-known Lazarević inequality 1, 2]statesthat sinh p > cosh 1.1 for all >0ifandonlyifp 3. Inequality 1.1wasgeneralizedbyZhu3] as follows. Let p, 8/15] 1,. Then the inequality sinh q > p +1 p cosh holds for >0ifandonlyifq 31 p. For p >0, Yang4] proved that the inequality sinh 3p 2 > coshp 1.2 holds for all >0ifandonlyifp 5/5, and inequality 1.2 isreversedifandonlyif p 1/3. Neuman and Sándor 5] proved that the Cusa type inequality sinh 2+cosh holds for all > Yang and Chu. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original authors and the source, provide a link to the Creative Commons license, and indicate if changes were made.

2 Yangand ChuJournal of Inequalities and Applications :403 Page 2 of 19 In 6], Zhu presented a more general result which contains Lazarević inequality 1.1and Cusa type inequality 1.3 as follows. Theorem A The following statements are true: i If p 4/5, then the double inequality sinh p 1 λ + λ cosh p 1 η + η cosh p holds for all >0if and only if λ 0 and η 1/3. ii If p 0, then the inequality sinh p 1 η + η cosh p holds for all >0if and only if η 1/3. More inequalities for the hyperbolic sine and cosine functions can be found in the literature 7 19]. Let p, q R and H p,q bedefinedon0, by H p,q U p sinh U q cosh, 1.4 where the function U p tisdefinedon1, by U p t tp 1 p t p 1 p 0, U 0 tlim log t. 1.5 p 0 p The main purpose of this paper is to deal with the monotonicity of H p,q on0,, generalize and improve the Lazarević and Cusa type inequalities, and present the new bounds for certain bivariate means. 2 Monotonicity Lemma 2.1 Let p R and U p t be defined on 1, by 1.5. Then the function p U p t is increasing on R and U p t>0for all t 1,. Proof Let p 0, then the monotonicity of the function p U p t follows easily from U p t p tp p 2 t p 1 log t p] >0. It follows from the monotonicity of the function p U p tthat U p t> lim U t p 1 pt lim 0. p p p Lemma 2.2 See 11, 20] Let f, g :a, b] R be continuous on a, b] and differentiable on a, b, and g 0on a, b. If f /g is increasing decreasing on a, b, then the functions f f a g ga, f f b g gb are also increasing decreasing on a, b.

3 Yangand ChuJournal of Inequalities and Applications :403 Page 3 of 19 Lemma 2.3 See 21] Let a n } n0 and b n} n0 be two real sequences with b n >0for n 0,1,2,..., and the power series At n0 a nt n and Bt n0 b nt n have the radius of convergence r >0.If the non-constant sequence a n /b n } n0 is increasing decreasing, then the function At/Bt is also increasing decreasing on 0, r. Let Sh p andch p bedefinedon0, by sinh Sh p U p 2.1 and Ch p U p cosh, 2.2 respectively, where the function U p is defined by 1.5. Then from 1.4and1.5 we clearly see that the function H p,q canberewrittenas H p,q Sh p Ch p Sh p Sh p 0 + Ch p Ch p 0 + q sinh p 1 p cosh q 1 1 sinh p 1 p logcosh sinh log, pq 0,, p 0,q 0, q cosh q, p 0,q 0, 1 log sinh logcosh, p q Let pq 0. Then it follows from 1.5, 2.1, and 2.2that where Sh p Ch q cosh1 q sinh p 1 ] cosh sinh : f1, 2 sinh 2.4 f 1 f 2 sinh p 2 sinh 3 cosh q, 2.5 f 2 pa qb+c 2.6 with A cosh sinh ] 2 cosh>0, 2.7 B cosh sinh ] sinh 2 >0, 2.8 C 2 2 cosh+ sinh+cosh sinh 2 >0, 2.9 where C>0dueto sinh C 2 2 cosh + tanh ] 2 >0 2 by the Wilker type inequality given in 15].

4 Yangand ChuJournal of Inequalities and Applications :403 Page 4 of 19 It is not difficult to verify that arealsotrueforpq 0. Let a n 4n 2 14n +9 9 n 1 +12n 2 10n 1, 2.10 b n 4nn 29 n 1 4nn 2, 2.11 c n 9 n 32n 2 +24n Then making use of together with the power series formulas sinh n0 2n+1 /2n +1!andcosh n0 2n /2n! we have A cosh sinh ] 2 cosh cosh cosh 1 2 sinh3 1 2 sinh+ 1 4 cosh3 1 4 cosh n3 4n 2 14n +99 n 1 +12n 2 10n 1 2n 42n! B cosh sinh ] sinh 2 n cosh cosh 1 4 sinh sinh n3 4nn 29 n 1 4nn 2 2n 42n! n3 C 2 2 cosh+ sinh+cosh sinh cosh+ sinh+ 1 4 cosh3 1 4 cosh n3 9 n 32n 2 +24n 1 2n 42n! n3 a n 42n! 2n, 2.13 b n 42n! 2n, 2.14 c n 42n! 2n In order to investigate the monotonicity of the function H p,q, we need Lemma 2.4. Lemma 2.4 Let A, B, and C be, respectively, defined by 2.7, 2.8, and 2.9, f 3 be defined on 0, by f 3 pa qb +1, 2.16 C I 1 q 0,p >0} q >0, p q 23 } q 0, pq } 17 1, 2.17 and I 2 q 0,p 0} q >0, pq } 1 q 0, p q 23 } Then the following statements are true:

5 Yangand ChuJournal of Inequalities and Applications :403 Page 5 of 19 i f 3 is increasing from 0, onto 5p 15q/8 + 1, if p, q I 1 ; ii f 3 is decreasing from 0, onto,5p 15q/8 + 1 if p, q I 2. Proof Let a n, b n,andc n be, respectively, defined by 2.10, 2.11, and Then it follows from that f 3 1 pa n qb n n3 2n 42n! c n n3 42n! 2n, 2.19 pa n qb n 42n! c n 42n! pa n qb n c n, 2.20 pa n+1 qb n+1 c n+1 pv n qu n c n c n+1 pa n qb n c n pa n+1c n a n c n+1 qb n+1 c n b n c n+1 c n c n+1 p v n c n c n+1, q 0, v n c n c n+1 q p q u n v n, q 0, 2.21 where u n b n+1 c n b n c n n 1 36n 189 n 512n 4 384n 3 560n n 36 ] +4 42n +40n 2 1, 2.22 v n a n+1 c n a n c n n 1 36n 459 n 512n 4 1,152n 3 +1,072n 2] n +59 n 16n 2 +60n +5 ] We claim that c n > and v n > for n 3. Indeed, making use of the binomial epansion we have c n 9 n 32n 2 +24n 11+8 n 32n 2 +24n 1 nn 1 >1+8n n 2 +24n 10, 2 36n 459 n 512n 4 1,152n 3 +1,072n 2 >36n n + nn nn 1n

6 Yangand ChuJournal of Inequalities and Applications :403 Page 6 of n 4 1,152n 3 +1,072n 2 2,560n ,968n ,760n ,340n 3+25,991>0, 228n +59 n 16n 2 +60n +5 >228n n 16n 2 +60n n 2 +76n +5>0 for n 3. We divide the proof into two cases. Case 1. q 0. Then , 2.24, 2.25, and Lemma 2.3 lead to the conclusion that f 3 isincreasingon0, ifp >0anddecreasingon0, ifp 0. Therefore, we have 5p/8 + 1 lim 0 + f 3 f 3 lim f 3 for p >0,f 3 1forp 0,and lim f 3 f 3 lim 0 + f 3 5p/8 + 1 for p 0. Case 2. q 0.Wefirstprovethatu n /v n is decreasing for n 3. From 2.25 weknow that it suffices to show that u n v n+1 u n+1 v n >0forn 3. It follows from 2.22and2.23 that u n v n+1 u n+1 v n 16c n+1 9 3n+2 1,024n 4 2,560n 3 +2,752n n 3 + 1,024n 4 +2,560n 3 +2,752n n 81 ] Note that 9 n+2 1,024n 4 2,560n 3 +2,752n >1+8n +2+ n +2n n +2n +1n n +2n +1nn n +2n +1nn 1n ,024n 4 2,560n 3 +2,752n ,048n ,320n ,680n ,040n ,692n ,605 > for n 3. Therefore, u n v n+1 u n+1 v n >0forn 3 follows from 2.24, 2.26, and From 2.21, 2.24, 2.25, and the monotonicity of u n /v n we clearly see that u n 1 lim u n u 3 23 n v n v n v 3 17, pa n+1 qb n+1 c n+1 pa n qb n c n >0, q >0, p q 23 17, 0, q 0, p q 23 17, 0, q >0, p q 1, >0, q 0, p q

7 Yangand ChuJournal of Inequalities and Applications :403 Page 7 of 19 Therefore, the desired results follows easily from 2.19, 2.20, 2.28, and Lemma 2.3 together with the facts that lim f 5p 15q lim f 3 +1,, q >0, p q or q 0,p q 1,, q 0, p q or q >0,p q 1. From Lemma 2.4, we get the monotonicity of H p,q as follows. Proposition 2.1 Let I 1 and I 2 be defined, respectively, by 2.17 and 2.18, and H p,q be defined on 0, by 2.3.Then the following statements are true: i H p,q is increasing on 0, if p, q I 1 0, 0 5p 15q/ }; ii H p,q is decreasing on 0, if p, q I 2 5p 15q/ }. Proof Let f 2 andf 3 bedefinedby2.6and2.16, respectively. i If p, q I 1 0, 0 5p 15q/8+1 0},thenH p,q isincreasingon0, follows easily from , 2.9, 2.16, Lemma 2.2, and Lemma 2.4i. ii If p, q I 2 5p 15q/8+1 0},thenH p,q isdecreasingon0, follows easily from , 2.9, 2.16, Lemma 2.2, and Lemma 2.4ii. It is easily to verify that p, q I 1 0, 0 5p 15q/ } is equivalent to 3q 8/5, q 34/35,, p 23q/17, q 0, 34/35, q, q,0 or p/3 + 8/15, p 46/35,, q 17p/23, p 0, 46/35, p, p,0, and p, q I 2 5p 15q/ } is equivalent to p q, q 4/5,, 3q 8/5, q, 4/5 or q p, p 4/5,, p/3 + 8/15, p,4/5. Therefore, Proposition 2.1 can be restated as Propositions 2.2 and 2.3. Proposition 2.2 Let H p,q be defined on 0, by 2.3. Then the following statements are true: i If q 34/35,, then H p,q is increasing on 0, for p 3q 8/5and decreasing for p q.

8 Yangand ChuJournal of Inequalities and Applications :403 Page 8 of 19 ii If q 4/5, 34/35, then H p,q is increasing on 0, for p 23q/17 and decreasing for p q. iii If q 0, 4/5, then H p,q is increasing on 0, for p 23q/17 and decreasing for p 3q 8/5. iv If q,0], then H p,q is increasing on 0, for p q and decreasing for p 3q 8/5. Proposition 2.3 Let H p,q be defined on 0, by 2.3. Then the following statements are true: i If p 46/35,, then H p,q is increasing on 0, for q p/3 + 8/15 and decreasing for q p. ii If p 4/5, 46/35, then H p,q is increasing on 0, for q 17p/23 and decreasing for q p. iii If p 0, 4/5, then H p,q is increasing on 0, for q 17p/23 and decreasing for q p/3 + 8/15. iv If p,0], then H p,q is increasing on 0, for q p and decreasing for q p/3 + 8/15. Let p kq,thenproposition2.1 leads to the following corollary. Corollary 2.1 Let H p,q be defined on 0, by 2.3. Then the following statements are true: i If k 3,, then H kq,q is increasing on 0, for q 0 and decreasing for q 8/53 k]. ii If k 3, then H kq,q is increasing on 0, for all q R. iii If k 23/17, 3, then H kq,q is increasing on 0, for 0 q 8/53 k]. iv If k 1, 23/17, then H kq,q is increasing on 0, for q 0. v If k,1], then H kq,q is increasing on 0, for q 0 and decreasing for q 8/53 k]. Let I 1 and I 2 be defined by 2.17and2.18, respectively. If 5p 15q/ , then we clearly see that 5p 15q I 1 0, 0} 8 5p 15q I } } +10 q 34 q 4 5, p 3q , p 3q 8 5 } and Proposition 2.1 leads to the followingcorollary. } p 4 5p +8, q 5 15 p 46 5p +8, q Corollary 2.2 Let H p,q be defined on 0, by 2.3. Then H 3q 8/5,q is increasing on 0, if q 34/35 and decreasing if q 4/5. In other words, H p,5p+8/15 is increasing on 0, if p 46/35 and decreasing if p 4/5. }, }, 3 Main results In this section, we present several Lazarević and Cusa type inequalities involving the hyperbolic sine and cosine functions with two parameters.

9 Yangand ChuJournal of Inequalities and Applications :403 Page 9 of 19 Let Sh p, Ch p, H p,q, I 1,andI 2 be, respectively, defined by 2.1, 2.2, 2.3, 2.17, and Then it is not difficult to verify that 5p 15q I 1 0, 0} 8 0 q min 5p 15q I 2 8 ma 17p H p,q } p 5p +8, } +1 0 } q min 0, p, } 5p +8, q 0 q ma 0, p, 15 } 5p +8, 15 } 5p +8, 15 From Proposition 2.1,we get Theorem 3.1 immediately. Theorem 3.1 i If 0 q min17p/23, 5p + 8/15} or q min0, p,5p + 8/15}, then the inequalities sinh p 1 p sinh log > 1 cosh q 1 3 q > 1 cosh q 1 3 q pq 0, 3.1 p 0,q 0, 3.2 sinh p 1 > 1 p 3 log cosh ] p 0,q 0, 3.3 sinh log > 1 3 log cosh ] p q 0, 3.4 hold for 0, with the best possible constant 1/3. ii If ma17p/23, 5p + 8/15} q 0or q ma0, p,5p + 8/15}, then all the inequalities are reversed. For clarity of epression, in the following we directly write Sh p, Ch p, and H p,q, and so on. For their general formulas, if pq 0, then we regard them as limit at p 0or q 0, unless otherwise specified. Lemma 3.1 will be used to establish sharp inequalities for hyperbolic functions. Lemma 3.1 Let Sh p and Ch q be, respectively, defined on 0, by 2.1 and 2.2, and D p,q be defined on 0, by D p,q Sh p 1 3 Ch q sinh p 1 p coshq q Then we have D p,q lim 1 p 3q + 8,

10 Yangand ChuJournal of Inequalities and Applications :403 Page 10 of 19 D 3q 8/5,q lim 1 q 34, , p > q 0, lim e q D p,q ], p q 0, 2 q 3q, q p >0, 2 q 3q, q > p 0, 3.8, p 0, q 0, lim D p,q, p 0,q 0, q 1 p, p 0,q 0. Proof Let 0 +, then making use of power series formulas and 3.5weget D p,q 5p 15q p2 42p 315q q o ,360 Therefore, 3.6and3.7follows easily from We divide the proof of 3.8intofourcases. Case 1. p >0,q >0.Then3.5leadsto lim e q D p,q ] 1 e p q 1 e 2 p lim 1 p p 2 3q, p > q >0, 2 q 3q, q p >0. 1+e 2 Case 2. p 0,q > 0. Then it follows from 3.5that 2 q 1 p 1 ] e q 3q lim e q D 0,q ] 1 e lim e q + e q 2 log e q log 1 1+e 2 q ] + e q 2 3q 2 3q 2 q 3q. Case 3. p 0,q 0.Then3.5leadsto 2 lim D 0,0 lim 3 log 1 e 2 ] + log 1+e Case 4. p >0,q 0. Then it follows from Lemma 2.1 and 3.5that D p,0 >D 0, for 0,. Therefore, lim D p,0 follows from Case 3 and 3.11.

11 Yangand ChuJournal of Inequalities and Applications :403 Page 11 of 19 Equation 3.9 followseasily from 3.5and the fact that lim U p, p 0, 1 p, p 0. Making use of Proposition 2.2 and Lemma 3.1 we get Theorems 3.2 and 3.3. Theorem 3.2 The following statements are true: i If q 34/35,, then the double inequality sinh p 2 1 coshq 1 p 2 3q sinh p 1 1 p holds for 0, if and only if p 1 3q 8/5and p 2 q. ii If q 4/5, 34/25, then the second inequality of 3.12 holds for 0, if p 1 23q/17, and the first inequality of 3.12 holds for 0, if and only if p 2 q. iii If q 0, 4/5, then the second inequality of 3.12 holds for 0, if p 1 23q/17, and the first inequality of 3.12 holds for 0, if and only if p 2 3q 8/5. iv If q,0], then the second inequality of 3.12 holds for 0, if p 1 q, and the first inequality of 3.12 holds for 0, if and only and p 2 3q 8/5. Proof All the sufficiencies in i-iv follow from Proposition 2.2 and H p,q 0 + 1/3. Net, we prove the necessities in i-iv. i If q 34/35, and the second inequality of 3.12holdsforall 0,, then 3.5 and 3.6leadtotheconclusionthatp 1 3q 8/5. If q 34/35, and the first inequality of 3.12holdsforall 0,, then we claim that p 2 q,otherwise,p 2 > q 34/35, and the first inequality 3.12 implythatd p2,q 0 for all 0,, which contradicts with 3.8. ii If q 4/5, 34/25 and the first inequality of 3.12holdsfor 0,, then the proof of p 2 q issimilartoparttwoofi. iii If q 0, 4/5 and the first inequality of 3.12holdsfor 0,, then the proof of p 2 3q 8/5issimilartopartoneofi. iv If q, 0] and the first inequality of 3.12 holdsfor 0,, then 3.5 and 3.6leadtotheconclusionthatp 2 3q 8/5. Theorem 3.3 The following statements are true: i If p 46/35,, then the double inequality cosh q 1 1 3q 1 sinh p 1 coshq p 3q 2 holds for all 0, if and only if q 1 5p + 8/15 and q 2 p. ii If p 4/5, 46/35, then the first inequality of 3.13 holds for all 0, if q 1 17p/23, and the second inequality of 3.13 holds for all 0, if and only if q 2 p.

12 Yangand ChuJournal of Inequalities and Applications :403 Page 12 of 19 iii If p 0, 4/5, then the first inequality of 3.13 holds for all 0, if q 1 17p/23, and the second inequality of 3.13 holds for all 0, if and only if q 2 5p + 8/15. iv If p,0], then the first inequality of 3.13 holds for all 0, if q 1 p, and the second inequality of 3.13 holds for all 0, if and only if q 2 5p + 8/15. Remark 3.1 Let q 1. Then Theorem 3.2i leads to the conclusion that the inequality sinh 31 p > p +1 p cosh 3.14 holds for all 0, ifandonlyifp 8/15, and inequality 3.14isreversedifandonly if p 2/3. Remark 3.2 Let t 1,, and p,q and Mt; p, q be, respectively, defined by p,q p, q:p 0 } p, q:3q p 0 } 3.15 and 1 p 3q + p 3q tq 1/p, pq 0,p, q p,q, e Mt; p, q tq 1/3q, p 0,q 0, p 3 log t +11/p, p >0,q 0, t 1/3, p q Then we clearly see that H p,q >H p,q 0 + 1/3 for all 0, isequivalentto sinh/ > Mcosh; p, q. Moreover, it is not difficult to verify that Mt; p, q isde- creasing with respect to p andincreasingwithrespecttoq if p, q p,q. Remark 3.3 From Remark 3.2 we know that Theorems 3.2 and 3.3 arealsotrueifp, q p,q and replacing 3.12and3.13, respectively, with 1 p 1 3q + p 1/p1 1 3q coshq sinh 1 p 2 3q + p 1/p2 2 3q coshq and 1 p + p 1/p cosh q 1 sinh 1 p + p 1/p cosh q 2. 3q 1 3q 1 3q 2 3q 2 Let q 1, then Theorem 3.2 leads to Corollary 3.1. Corollary 3.1 Thedoubleinequality 1 p p 1/p1 1 3 cosh sinh 1 p p 1/p2 2 3 cosh holds for all 0, if and only if p 1 7/5 and 0 p 2 1.

13 Yangand ChuJournal of Inequalities and Applications :403 Page 13 of 19 Remark 3.4 Letting p 1 7/5, 3/2, 2, 3 and making use of the monotonicity of Mcosh; p, qwithrespecttop, then Corollary 3.1 leads to the inequalities 1 cosh 1/ /2 1 3 cosh /3 2 cosh /7 15 cosh sinh cosh for 0,, which is better than the inequalities given in 3.23 of 7]. Let p 0, 1, then Theorem 3.3 leads to Corollary 3.2. Corollary 3.2 Let 0,. Then the double inequality e coshq 1 1/3q 1 sinh e coshq 2 1/3q 2 holds if and only if q 1 0 and q 2 8/5, and the double inequality cosh q 1 sinh cosh q q 1 3q 1 3q 2 3q 2 holds if and only if q 1 17/23 and q 2 1. Remark 3.5 Letting q 1 17/23, 2/3, 1/2, 1/3, 1/6, 0 and making use of the monotonicity of Mcosh;p, q withrespecttoq, theninequality3.17 leadsto 1 3 log cosh ] +12cosh 1/6 1cosh 1/ cosh1/ cosh2/ cosh17/23 sinh cosh for 0,. Let p kq,then3.15and3.16become kq,q k, q:k, q 0 } k, q:k, q 0 } k, q:0k 3, q 0 }, 1 k 3 + k 3 tq 1/kqt, kq 0,k, q kq,q, Mt; kq, q e tq 1 3q, q 0,k 0, t 1/3, q 0. Remark 3.6 It is not difficult to verify that Mt; kq, q is decreasing increasing with respect to q if k >3,andMt; kq, q is decreasing increasing with respect to k if q >0. Theorem 3.4 Let 0, and k 0,3.Then the following statements are true:

14 Yangand ChuJournal of Inequalities and Applications :403 Page 14 of 19 i If k 23/17, 3, then the inequality sinh > 1 k 3 + k 1/kq 3 coshq 3.18 holds if and only if q 8/53 k]. ii If k 0, 1], then the double inequality 1 k 3 + k 1/kq1 3 coshq 1 sinh 1 k 3 + k 1/kq2 3 coshq holds if and only if q 1 0 and q 2 8/53 k]. Proof i If k 23/17, 3, then it follows from Corollary 2.1iii that H kq,q >H kq,q 0 + 1/3 and 3.18 holdsfor0 q 8/53 k]. For q 0,wehaveM k cosh; kq, q M k cosh; 0, 0 due to M k cosh; kq, q isaweightedqth power mean of cosh and 1, so 3.18 still holds. If 3.18holds,thenD kq,q >0and3.6leadsto D kq,q lim 1 kq 3q and q 8/53 k]. ii If k 0, 1], then it follows from Corollary 2.1v that H kq1,q 1 >H kq1,q /3, H kq2,q 2 H kq2,q /3, and 3.19holdsforq 1 0andq 8/53 k]. If the second inequality of 3.19holds,thenD kq2,q 2 0and3.6leadstoq 2 8/53 k]. Net, we prove that the condition q 1 0 is necessary such that the first inequality of 3.19holdsforall 0,. Indeed, the first inequality of 3.19leadstoD kq1,q 1 0for all 0,. If q 1 >0andk 0, 1], then 0 kq 1 q 1 and 3.8leadsto lim e q 1 D kq1,q 1 ] 2 q 1 0, 3q 1 which implies that there eists large enough X >0suchthatD kq1,q 1 0for X,. Let k 1,3/2,2,thenTheorem3.4 leads to Corollary 3.3. Corollary 3.3 The inequalities and /p1 3 coshp 1 sinh sinh sinh 1 > coshp /p2 3 coshp 2, /3p, > /2q 3 coshq 3.22 hold for all 0, if and only if p 1 0, p 2 4/5, p 16/15, and q 8/5.

15 Yangand ChuJournal of Inequalities and Applications :403 Page 15 of 19 Let p 3q 8/5,then3.15and3.16become 3q 8/5,q q : q 8 }, 15 M t;3q 85 8, q 15q q tq 5/15q 8, q > 8 15, e 5t8/15 1/8, q It is easy to prove that Mt;3q 8/5, q is decreasing with respect to q on the interval 8/15, and lim t;3q M 85, q t 1/3. q Theorem 3.5 Let q > 8/15. Then the inequality sinh 8 > 15q /15q 8 cosh ] q q holds for all 0, if and only if q 34/35, and inequality 3.23 is reversed if and only if q 4/5. Proof The sufficiency can be derived from Corollary 2.2. Ifinequality3.23 holds,then D 3q 8/5,q >0and3.7 leads to the conclusion that q 34/35. Net, we prove that q 4/5 if the reversed inequality 3.23holds. If there eists q > 4/5 such that the reversed inequality 3.23holds,thenD 3q 8/5,q 0, 3q 8/5 > q > 4/5, and 3.8leadsto lim e q D 3q 8/5,q ], 3.24 which contradicts with D 3q 8/5,q 0. Let q 34/35,1,16/15,6/5,8/5,2, and 4/5, 7/10, 2/3, 3/5, 8/15 +. Then Theorem 3.5 leads to Corollary 3.4. Corollary 3.4 The inequalities 11 cosh 1/ cosh / cosh8/ / cosh6/ / cosh16/ / cosh cosh34/ /46 sinh cosh4/ / /2 e 5cosh8/15 1/8 23 hold for all 0, cosh7/ cosh2/ /7

16 Yangand ChuJournal of Inequalities and Applications :403 Page 16 of 19 4 Applications Let a, b > 0. Then the geometric mean Ga, b, arithmetic mean Aa, b, quadratic mean Qa, b, logarithmic mean La, b, Neuman-Sándor mean Ma, b 22] and second Yang mean Va, b23]aredefinedby Ga, b ab, Aa, b a + b a2 + b, Qa, b 2, 2 2 and La, b b a log b log a a b, La, aa, b a Ma, b 2 sinh 1 b a b+a a b, Ma, aa, Va, b b a Va, b a b, Va, aa, 2 sinh 1 b a 2ab respectively. The Schwab-Borchardt mean SBa, b22, 24, 25]ofa 0andb >0isgivenby b 2 a 2 arccosa/b, a b, SBa, b a, a b, a 2 b 2, a > b, cosh 1 a/b where cosh 1 log is the inverse hyperbolic cosine function. Let p, q R and the function t Shp, q, tbedefinedon0, by q sinhpt p sinhqt 1/p q, pqp q 0, sinhpt 1/p, p 0,q 0, pt Shp, q, t sinhqt 1/q, p 0,q 0, qt e t cothpt 1/p, p q, pq 0, 1, p q 0. Recently, Yang 23]provedthatSh p,q b, a, defined by Sh p,q b, a a Shp, q, cosh 1 b/a], a b, a, a b, is a mean of a and b for all b a >0ifp, q p, q :p >0,q >0,p + q 3, Lp, q 1/ log 2} p, q:p 0,0 p + q 3} p, q:q 0,0 p + q 3}. In particular, Sh 1,0 b, aasinhcosh 1 b/a]/ cosh 1 b/asbb, a for all b a >0.Let t cosh 1 b/a, then Theorems lead to Theorems Theorem 4.1 Let b a >0and p, q p, q :p 0} p, q :3q p 0}. Then the following statements are true:

17 Yangand ChuJournal of Inequalities and Applications :403 Page 17 of 19 i If q 34/35,, then the double inequality 1 p 1 a q + p 1 3q SBb, a 3q bq 1 p 2 3q ] 1/p1 a 1 q/p 1 a q + p ] 1/p2 2 3q bq a 1 q/p holds if and only if p 1 3q 8/5and p 2 q. ii If q 4/5, 34/35, then the first inequality of 4.1 holds for p 1 23q/17, and the second inequality of 4.1 holds if and only if p 2 q. iii If q 0, 4/5, then the first inequality of 4.1 holds for p 1 23q/17, and the second inequality of 4.1 holds if and only if p 2 3q 8/5. iv If q,0], then the first inequality of 4.1 holds for p 1 q, and the second inequality of 4.1 holds if and only if p 2 3q 8/5. Theorem 4.2 Let b a >0and p, q p, q :p 0} p, q :3q p 0}. Then the following statements are true: i If p 46/35,, then the double inequality 1 p a q 1 + p 3q 1 SBb, a b q 1 3q 1 1 p 3q 2 ] 1/p a 1 q 1/p a q 2 + p ] 1/p b q 1 a 1 q 2/p 3q holds if and only if q 1 5p + 8/15 and q 2 p. ii If p 4/5, 46/35, then the first inequality of 4.2 holds for q 1 17p/23, and the second inequality of 4.2 holds if and only if q 2 p. iii If p 0, 4/5, then the first inequality of 4.2 holds for q 1 17p/23, and the second inequality of 4.2 holds if and only if q 2 5p + 8/15. iv If p,0], then the first inequality of 4.2 holds for q 1 p, and the second inequality of 4.2 holds if and only if q 2 5p + 8/15. Theorem 4.3 Let b a >0and k 0, 3. Then the following statements are true: i If k 23/17, 3, then the inequality SBb, a> 1 k a q + k ] 1/kq 3 3 bq a 1 1/k holds if and only if q 8/53 k]. ii If k 0, 1], then the double inequality 1 k a q 1 + k ] 1/kq1 3 3 bq 1 a 1 1/k SBb, a 1 k a q 2 + k ] 1/kq2 3 3 bq 2 a 1 1/k holds if and only if q 1 0 and q 2 8/53 k].

18 Yangand ChuJournal of Inequalities and Applications :403 Page 18 of 19 Theorem 4.4 Let b a >0and p, q > 8/15. Then the double inequality 8 15p ap p SBb, a b p ] 5/15p 8 a p/15p q aq + holds if and only if p 34/35 and q 4/ ] 5/15q 8 b q a q/15q 8 15q Remark 4.1 Let a, b >0witha b. Then we clearly see that Sh 1,0 Aa, b, Ga, b ] SB Aa, b, Ga, b ] b a La, b, log b log a ] ] b a Sh 1,0 Qa, b, Aa, b SB Qa, b, Aa, b 2 sinh 1 b a Ma, b, b+a Sh 1,0 Qa, b, Ga, b ] SB Qa, b, Ga, b ] b a Va, b, 2 sinh 1 b a 2ab and Theorems still hold true if we replace b, a, SBa, b with Aa, b, Ga, b, La, b, Qa, b, Aa, b, Ma, b and Qa, b, Ga, b, Va, b. Competing interests The authors declare that they have no competing interests. Authors contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Author details 1 School of Mathematics and Computation Sciences, Hunan City University, Yiyang, , China. 2 Department of Mathematics, Huzhou University, Huzhou, , China. Acknowledgements The research was supported by the Natural Science Foundation of China under Grant and the Natural Science Foundation of Zhejiang Province under Grant LY13A Received: 28 August 2015 Accepted: 1 December 2015 References 1. Lazarević, H: Certain inequalities with hyperbolic functions. Publ. Elektroteh. Fak. Univ. Beogr., Ser. Mat. Fiz , Mitrinović, DS: Analytic Inequalities. Springer, New York Zhu, L: Generalized Lazarevic s inequality and its applications II. J. Inequal. Appl. 2009, Article ID Yang, Z-H: New sharp bounds for logarithmic mean and identric mean. J. Inequal. Appl. 2013, Article ID Neuman, E, Sándor, J: On some inequalities involving trigonometric and hyperbolic functions with emphasis on the Cusa-Huygens, Wilker, and Huygens inequalities. Math. Inequal. Appl. 134, Zhu, L: Inequalities for hyperbolic functions and their applications. J. Inequal. Appl. 2010, Article ID Chen, C-P, Sándor, J: Inequality chains for Wilker, Huygens and Lazarević type inequalities. J. Math. Inequal. 81, Neuman, E: Refinements and generalizations of certain inequalities involving trigonometric and hyperbolic functions. Adv. Inequal. Appl. 11, Sándor, J: Two sharp inequalities for trigonometric and hyperbolic functions. Math. Inequal. Appl. 152, Sándor, J: Trigonometric and hyperbolic inequalities arxiv: math.ca] 11. Vamanamurthy, MK, Vuorinen, M: Inequalities for means. J. Math. Anal. Appl. 1831, Wu, S-H, Debnath, L: Wilker-type inequalities for hyperbolic functions. Appl. Math. Lett. 255, Yang, Z-H: New sharp bounds for identric mean in terms of logarithmic mean and arithmetic mean. J. Math. Inequal. 64, Yang, Z-H, Chu, Y-M: A note on Jordan, Adamović-Mitrinović, and Cusa inequalities. Abstr. Appl. Anal. 2014, Article ID Zhu, L: On Wilker-type inequalities. Math. Inequal. Appl. 104,

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