Li, Wen-Hui

Qi, Feng

Repository of the Academy's Library

Miskolc Mathematical Notes HU e-ISSN 1787-2413Vol. 15 (2014), No 2, pp. 665-675 DOI: 10.18514/MMN.2014.1176A uniﬁed proof of several inequalities andsome new inequalities involvingNeuman-Sándor meanFeng Qi and Wen-Hui LiMiskolc Mathematical Notes HU e-ISSN 1787-2413Vol. 15 (2014), No. 2, pp. 665–675A UNIFIED PROOF OF SEVERAL INEQUALITIES AND SOMENEW INEQUALITIES INVOLVING NEUMAN-SA´NDOR MEANFENG QI AND WEN-HUI LIReceived 21 March, 2014Abstract. In the paper, by finding linear relations of differences between some means, the authorssupply a unified proof of several double inequalities for bounding Neuman-Sa´ndor means interms of the arithmetic, harmonic, and contra-harmonic means and discover some new sharpinequalities involving Neuman-Sa´ndor, contra-harmonic, root-square, and other means of twopositive real numbers.2010 Mathematics Subject Classification: 26E60; 26D07; 41A30Keywords: mean, inequality, Neuman-Sa´ndor mean, monotonicity, unified proof, hyperbolicsine, hyperbolic cosine1. INTRODUCTIONIt is well known that the quantitiesA.a;b/D aCb2; G.a;b/Dpab ;H.a;b/D 2abaCb ; C .a;b/D2.a2CabCb2/3.aCb/ ;C.a;b/D a2Cb2aCb ; P.a;b/Da b4arctanpa=b   ;Q.a;b/Dsa2Cb22; T .a;b/D a b2arctan a baCbare respectively called in the literature the arithmetic, geometric, harmonic, centroidal,contra-harmonic, first Seiffert, root-square, and second Seiffert means of two positivereal numbers a and b with a¤ b.For a;b > 0 with a¤ b, Neuman-Sa´ndor mean M.a;b/ is defined in [11] byM.a;b/D a b2arcsinh a baCb;c 2014 Miskolc University Press666 F. QI AND W.-H. LIwhere arcsinhx D ln.xCpx2C1/ is the inverse hyperbolic sine function. At thesame time, a chain of inequalitiesG.a;b/ < L 1.a;b/ < P.a;b/ < A.a;b/ <M.a;b/ < T .a;b/ < Q.a;b/were given in [11], whereLp.a;b/D8ˆˆˆˆˆˆ<ˆˆˆˆˆˆ:bpC1 apC1.pC1/.b a/1=p; p ¤ 1;01ebbaa1=.b a/; p D 0b alnb  lna ; p D 1is the p-th generalized logarithmic mean of a and b with a¤ b.In [11, 12], it was established thatA.a;b/ <M.a;b/ < T .a;b/; P.a;b/ <M.a;b/ < T 2.a;b/;A.a;b/T .a;b/ <M 2.a;b/ <A2.a;b/CT 2.a;b/2for a;b > 0 with a¤ b.For 0 < a;b < 12with a¤ b, Ky Fan type inequalitiesG.a;b/G.1 a;1 b/ <L 1.a;b/L 1.1 a;1 b/ <P.a;b/P.1 a;1 b/<A.a;b/A.1 a;1 b/ <M.a;b/M.1 a;1 b/ <T .a;b/T .1 a;1 b/were presented in [11, Proposition 2.2].In [8], it was showed that the double inequalityLp0.a;b/ <M.a;b/ < L2.a;b/holds for all a;b > 0 with a ¤ b and for p0 D 1:843: : : , where p0 is the uniquesolution of the equation .pC1/1=p D 2 ln.1Cp2/.In [10], Neuman proved that the double inequalities˛Q.a;b/C .1 ˛/A.a;b/ <M.a;b/ < ˇQ.a;b/C .1 ˇ/A.a;b/andC.a;b/C .1 /A.a;b/ <M.a;b/ < C.a;b/C .1 /A.a;b/ (1.1)hold for all a;b > 0 with a¤ b if and only if˛  1  ln.1Cp2/.p2 1/ ln.1Cp2/ D 0:3249: : : ; ˇ 13SOME INEQUALITIES FOR NEUMAN-SA´NDOR MEAN 667and 1  ln.1Cp2/ln.1Cp2/ D 0:1345: : : ; 16:In [20, Theorems 1.1 to 1.3], it was found that the double inequalities˛1H.a;b/C .1 ˛1/Q.a;b/ <M.a;b/ < ˇ1H.a;b/C .1 ˇ1/Q.a;b/;˛2G.a;b/C .1 ˛2/Q.a;b/ <M.a;b/ < ˇ2G.a;b/C .1 ˇ2/Q.a;b/;and˛3H.a;b/C .1 ˛3/C.a;b/ <M.a;b/ < ˇ3H.a;b/C .1 ˇ3/C.a;b/ (1.2)hold for all a;b > 0 with a¤ b if and only if˛1  29D 0:2222: : : ; ˇ1  1  1p2 ln.1Cp2/ D 0:1977: : : ;˛2  13D 0:3333: : : ; ˇ2  1  1p2 ln.1Cp2/ D 0:1977: : : ;˛3  1  12 ln.1Cp2/ D 0:4327: : : ; ˇ3 512D 0:4166: : :In [19, Theorem 3.1], it was established that the double inequality˛I.a;b/C .1 ˛/Q.a;b/ <M.a;b/ < ˇI.a;b/C .1 ˇ/Q.a;b/holds for all a;b > 0 with a¤ b if and only if˛  12and ˇ  ep2 ln.1Cp2/ 1.p2e 2/ ln.1Cp2/ D 0:4121: : :For more information on this topic, please refer to [1–3,5,7–10,12–14,16–18,20]and plenty of references cited therein.The first goal of this paper is, by finding linear relations of differences betweensome means, to supply a unified proof of inequalities (1.1) and (1.2).The second purpose of this paper is to establish some new sharp inequalities in-volving Neuman-Sa´ndor, centroidal, contra-harmonic, and root-square means of twopositive real numbers a and b with a¤ b.2. LEMMASIn order to attain our aims, the following lemmas are needed.Lemma 1 ([15, Lemma 1.1]). Suppose that the power series f .x/ D1PnD0anxnand g.x/ D1PnD0bnxn have the radius of convergence r > 0 and bn > 0 for all n 2ND f0;1;2; : : :g. Let h.x/D f .x/g.x/. Then the following statements are true.668 F. QI AND W.-H. LI(1) If the sequence fanbng1nD0 is (strictly) increasing (decreasing), then h.x/ isalso (strictly) increasing (decreasing) on .0;r/.(2) If the sequence fanbng is (strictly) increasing (decreasing) for 0 < n  n0 and(strictly) decreasing (increasing) for n > n0, then there exists x0 2 .0;r/such that h.x/ is (strictly) increasing (decreasing) on .0;x0/ and (strictly)decreasing (increasing) on .x0; r/.Lemma 2. Leth1.x/D sinhx x2x sinh2x: (2.1)Then h1.x/ is strictly decreasing on .0;1/ and has the limit limx!0Ch1.x/D 112 .Proof. Let f1.x/D sinhx x and f2.x/D 2x sinh2x D x cosh2x x. Using thepower seriessinhx D1XnD0x2nC1.2nC1/Š and coshx D1XnD0x2n.2n/Š; (2.2)we can express the functions f1.x/ and f2.x/ asf1.x/D1XnD0x2nC3.2nC3/Š and f2.x/D1XnD022nC2x2nC3.2nC2/Š : (2.3)Hence, we haveh1.x/DP1nD0anx2nP1nD0 bnx2n; (2.4)where an D 1.2nC3/Š and bn D 22nC2.2nC2/Š . Let cn D anbn . Then cn D 1.2nC3/22nC2 andcnC1  cn D  .6nC17/.2nC3/.2nC5/22nC4 < 0:As a result, by Lemma 1, it follows that the function h1.x/ is strictly decreasing on.0;1/.From (2.4), it is easy to see that limx!0C h1.x/D a0b0 D 112 . The proof of Lemma 2is complete. Lemma 3. Leth2.x/D1  sinhxxC sinh2x3coshx  sinhxx: (2.5)Then h2.x/ is strictly increasing on .0;1/ and has the limit limx!0Ch2.x/D 12 .SOME INEQUALITIES FOR NEUMAN-SA´NDOR MEAN 669Proof. Letf3.x/D 1  sinhxxC sinh2x3D 1  sinhxxC cosh2x 16andf4.x/D coshx  sinhxx:Making use of the power series in (2.2) shows thatf3.x/D1XnD0.2nC3/22nC2 66.2nC3/Š x2nC2 and f4.x/D1XnD02nC2.2nC3/Šx2nC2:Therefore, we haveh2.x/DP1nD0anx2nC2P1nD0 bnx2nC2; (2.6)where an D .2nC3/22nC2 66.2nC3/Š and bn D 2nC2.2nC3/Š . Let cn D anbn . Thencn D .2nC3/22nC1 36.nC1/andcnC1  cn D 3C7 22nC2C21n 22nC1C3n2 22nC26.nC1/.nC2/ > 0:Accordingly, by Lemma 1, it follows that the function h2.x/ is strictly increasing on.0;1/.It is clear that limx!0C h2.x/D a0b0 D 12 . The proof of Lemma 3 is complete. Lemma 4. Leth3.x/Dcoshx  sinhxx1C sinh2x  sinhxx: (2.7)Then h3.x/ is strictly decreasing on .0;1/ and has the limit limx!0Ch3.x/D 25 .Proof. Letf5.x/D coshx  sinhxxandf6.x/D 1C sinh2x  sinhxxD 1  sinhxxC cosh2x 12:Utilizing the power series in (2.2) givesf5.x/D1XnD02nC2.2nC3/Šx2nC2 and f6.x/D1XnD0.2nC3/22nC1 1.2nC3/Š x2nC2:670 F. QI AND W.-H. LIThis implies thath3.x/DP1nD0anx2nC2P1nD0 bnx2nC2; (2.8)where an D 2nC2.2nC3/Š and bn D .2nC3/22nC1 1.2nC3/Š . Let cn D anbn . Thencn D 2nC2.2nC3/22nC1 1andcnC1  cn D  2.1C7 22nC2C21n 22nC1C3n2 22nC2/.3 22nC1Cn 22nC2 1/.5 22nC3Cn 22nC4 1/ < 0:In light of Lemma 1, we obtain that the function h3.x/ is strictly decreasing on.0;1/.It is obvious that limx!0C h3.x/D a0b0 D 25 . The proof of Lemma 4 is complete.3. A UNIFIED PROOF OF INEQUALITIES (1.1) AND (1.2)Now we are in a position to supply a unified proof of inequalities (1.1) and (1.2)and, as corollaries, to establish some new inequalities involving Neuman-Sa´ndor,contra-harmonic, centroidal, and root-square means of two positive real numbers aand b with a¤ b.It is not difficult to see that the inequalities (1.1) and (1.2) can be rearranged re-spectively as 1 < M.a;b/ C.a;b/C.a;b/ A.a;b/ <  1 (3.1)and ˛3 < M.a;b/ C.a;b/C.a;b/ H.a;b/ <  ˇ3: (3.2)The denominators in (3.1) and (3.2) meet2ŒC.a;b/ A.a;b/D C.a;b/ H.a;b/D .a b/2aCb (3.3)which were presented in [4, Eq. (4.4)]. This implies that the inequalities (1.1) and (1.2)are identical up to a scalar. Therefore, it is sufficient to prove one of the two inequal-ities (1.1) and (1.2).By a direct calculation, we also find6ŒC .a;b/ A.a;b/D 3ŒC.a;b/ C.a;b/D 2ŒA.a;b/ H.a;b/D 32ŒC .a;b/ H.a;b/D .a b/2aCb , CH.a;b/: (3.4)SOME INEQUALITIES FOR NEUMAN-SA´NDOR MEAN 671So, it is natural to raise a problem: what are the best constants ˛ and ˇ such that thedouble inequality˛ <M.a;b/ C.a;b/CH.a;b/< ˇ (3.5)holds for all a;b > 0 with a ¤ b? The following theorem gives a solution to thisproblem.Theorem 1. The double inequality (3.5) holds for all a;b > 0 with a ¤ b if andonly if˛  12 ln.1Cp2/  1D 0:4327: : : and ˇ   512D 0:4166: : :Proof. Without loss of generality, we assume that a > b > 0. Let x D ab. Thenx > 1 andM.a;b/ C.a;b/CH.a;b/Dx 12arcsinh x 1xC1  x2C1xC1.x 1/2xC1:Let t D x 1xC1 . Then t 2 .0;1/ andM.a;b/ C.a;b/CH.a;b/Dtarcsinh t   t2 12t2:Let t D sinh for  2  0; ln 1Cp2. ThenM.a;b/ C.a;b/CH.a;b/Dsinh  sinh2   12sinh2 D sinh  2 sinh2   12:In virtue of Lemma 2, Theorem 1 is thus proved. Corollary 1. The double inequality˛CH.a;b/CM.a;b/ < C.a;b/ < ˇCH.a;b/CM.a;b/ (3.6)holds for all a;b > 0 with a¤ b if and only if ˛  512D 0:4166: : : andˇ  1  12 ln.1Cp2/ D 0:4327: : :Corollary 2. The double inequality˛CH.a;b/CM.a;b/ < C.a;b/ < ˇCH.a;b/CM.a;b/ (3.7)holds for all a;b > 0 with a¤ b if and only if ˛  112D 0:0833: : : andˇ  23  12 ln.1Cp2/ D 0:0993: : :672 F. QI AND W.-H. LI4. SOME NEW INEQUALITIES INVOLVING NEUMAN-SA´NDOR MEANFinally we further establish some new inequalities involving Neuman-Sa´ndor, centroidal,root-square, and other means.Theorem 2. The inequalityM.a;b/ > CH.a;b/ (4.1)holds for all a;b > 0 with a¤ b if and only if  12 ln.1Cp2/ D 0:5672: : : .Proof. It is clear thatM.a;b/CH.a;b/D .a b/.aCb/.a b/22arcsinh a baCbD aCba b12arcsinh a baCb:Without loss of generality, we assume that a > b > 0. Let x D a baCb . Then x 2 .0;1/andM.a;b/CH.a;b/D 12x arcsinhx, f .x/:Differentiating f .x/ yieldsf 0.x/D xp1Cx2 C arcsinhx2x2 arcsinh2x 0which means that function f .x/ is decreasing for x 2 .0;1/.It is apparent thatlimx!1 f .x/D12 ln.1Cp2/ :The proof of Theorem 2 is thus complete. Theorem 3. The double inequality˛Q.a;b/C .1 ˛/M.a;b/ < C.a;b/ < ˇQ.a;b/C .1 ˇ/M.a;b/ (4.2)holds for all a;b > 0 with a¤ b if and only if ˛  12andˇ  3 4 ln.1Cp2/3Œ1 p2 ln.1Cp2/ D 0:7107: : :Proof. It is sufficient to show˛ <C.a;b/ M.a;b/Q.a;b/ M.a;b/ < ˇ:Without loss of generality, we assume that a > b > 0. Let x D ab. Then x > 1 andC.a;b/ M.a;b/Q.a;b/ M.a;b/ D2.x2CxC1/3.xC1/   x 12arcsinh x 1xC1qx2C12  x 12arcsinh x 1xC1:SOME INEQUALITIES FOR NEUMAN-SA´NDOR MEAN 673Let t D x 1xC1 . Then t 2 .0;1/ andC.a;b/ M.a;b/Q.a;b/ M.a;b/ Dt23C1  tarcsinh tp1C t2   tarcsinh t:Let t D sinh for  2  0; ln 1Cp2. ThenC.a;b/ M.a;b/Q.a;b/ M.a;b/ Dsinh2 3C1  sinhcosh   sinh:By Lemma 3, we obtain Theorem 3. Theorem 4. The double inequality˛C.a;b/C .1 ˛/M.a;b/ < Q.a;b/ < ˇC.a;b/C .1 ˇ/M.a;b/ (4.3)holds for all a;b > 0 with a¤ b if and only if˛ p2 ln.1Cp2/ 12 ln.1Cp2/ 1 D 0:3231: : : and ˇ 25:Proof. The double inequalities (4.3) is the same as˛ <Q.a;b/ M.a;b/C.a;b/ M.a;b/ < ˇ:Without loss of generality, we assume that a > b > 0. Let x D ab. Then x > 1 andQ.a;b/ M.a;b/C.a;b/ M.a;b/ Dqx2C12  x 12arcsinh x 1xC1x2C1xC1   x 12arcsinh x 1xC1:Let t D x 1xC1 . Then t 2 .0;1/ andQ.a;b/ M.a;b/C.a;b/ M.a;b/ Dp1C t2   tarcsinh t1C t2  tarcsinh t:Let t D sinh for  2  0; ln 1Cp2. ThenQ.a;b/ M.a;b/C.a;b/ M.a;b/ Dcosh   sinh1C sinh2    sinh:According to Lemma 4, the proof of Theorem 3 is complete. Remark 1. This paper is a slightly revised version of the preprint [6].ACKNOWLEDGEMENTSThe authors thank the anonymous referees for their careful corrections to the ori-ginal version of this paper.674 F. QI AND W.-H. LIREFERENCES[1] Y.-M. Chu and B.-Y. Long, “Bounds of the Neuman-Sa´ndor mean using power and identricmeans,” Abstr. Appl. Anal., pp. Art. ID 832 591, 6 pp., 2013.[2] Y.-M. Chu, B.-Y. Long, W.-M. Gong, and Y.-Q. Song, “Sharp bounds for Seiffert andNeuman-Sa´ndor means in terms of generalized logarithmic means,” J. Inequal. Appl., pp.2013:10, 13 pp., 2013. [Online]. Available: http://dx.doi.org/10.1186/1029-242X-2013-10[3] W.-D. Jiang and F. Qi, “Sharp bounds in terms of the power of the contra-harmonic mean forNeuman-Sa´ndor mean,” arXiv preprint. [Online]. Available: http://arxiv.org/abs/1301.3554[4] W.-D. Jiang and F. Qi, “Some sharp inequalities involving Seiffert and other means and theirconcise proofs,” Math. Inequal. Appl., vol. 15, no. 4, pp. 1007–1017, 2012. [Online]. Available:http://dx.doi.org/10.7153/mia-15-86[5] W.-D. Jiang and F. Qi, “Sharp bounds for Neuman-Sa´ndor’s mean in terms of theroot-mean-square,” Period. Math. Hungar., p. in press, 2014. [Online]. Available: http://dx.doi.org/10.1007/s10998-014-0057-9[6] W.-H. Li and F. Qi, “A unified proof of inequalities and some new inequalities involvingNeuman-Sa´ndor mean,” arXiv preprint. [Online]. Available: http://arxiv.org/abs/1312.3500[7] W.-H. Li and M.-M. Zheng, “Some inequalities for bounding Toader mean,” J. Funct. SpacesAppl., pp. Art. ID 394 194, 5 pp., 2013.[8] Y.-M. Li, B.-Y. Long, and Y.-M. Chu, “Sharp bounds for the Neuman-Sa´ndor mean in terms ofgeneralized logarithmic mean,” J. Math. Inequal., vol. 6, no. 4, pp. 567–577, 2012. [Online].Available: http://dx.doi.org/10.7153/jmi-06-54[9] E. Neuman, “Inequalities for the Schwab-Borchardt mean and their applications,” J. Math.Inequal., vol. 5, no. 4, pp. 601–609, 2011. [Online]. Available: http://dx.doi.org/10.7153/jmi-05-52[10] E. Neuman, “A note on a certain bivariate mean,” J. Math. Inequal., vol. 6, no. 4, pp. 637–643,2012. [Online]. Available: http://dx.doi.org/10.7153/jmi-06-62[11] E. Neuman and J. Sa´ndor, “On the Schwab-Borchardt mean,” Math. Pannon., vol. 14, no. 2, pp.253–266, 2003.[12] E. Neuman and J. Sa´ndor, “On the Schwab-Borchardt mean. II,” Math. Pannon., vol. 17, no. 1,pp. 49–59, 2006.[13] E. Neuman and J. Sa´ndor, “Companion inequalities for certain bivariate means,” Appl.Anal. Discrete Math., vol. 3, no. 1, pp. 46–51, 2009. [Online]. Available: http://dx.doi.org/10.2298/AADM0901046N[14] X.-H. Shen and Y.-M. Chu, “Bounds improvement for Neuman-Sa´ndor mean using arithmetic,quadratic and contraharmonic means,” Int. Math. Forum, vol. 8, no. 29-32, pp. 1477–1485, 2013.[15] S. Simic´ and M. Vuorinen, “Landen inequalities for zero-balanced hypergeometric functions,”Abstr. Appl. Anal., pp. Art. ID 932 061, 11 pp., 2012. [Online]. Available: http://dx.doi.org/10.1155/2012/932061[16] H. Sun, X.-H. Shen, T.-H. Zhao, and Y.-M. Chu, “Optimal bounds for the Neuman-Sa´ndor meansin terms of geometric and contraharmonic means,” Appl. Math. Sci. (Ruse), vol. 7, no. 85-88, pp.4363–4373, 2013. [Online]. Available: http://dx.doi.org/10.12988/ams.2013.35247[17] H. Sun, T. Zhao, Y. Chu, and B. Liu, “A note on the Neuman-Sa´ndor mean,” J. Math. Inequal.,vol. 8, no. 2, pp. 287–297, 2014. [Online]. Available: http://dx.doi.org/10.7153/jmi-08-20[18] M.-K. Wang, Y.-M. Chu, and B.-Y. Liu, “Sharp inequalities for the Neuman-Sa´ndor mean in termsof arithmetic and contra-harmonic means,” Rev. Anal. Nume´r. The´or. Approx., vol. 42, no. 2, pp.115–120, 2013.SOME INEQUALITIES FOR NEUMAN-SA´NDOR MEAN 675[19] T.-H. Zhao and Y.-M. Chu, “A sharp double inequality involving identric, Neuman-Sa´ndor, andquadratic means,” Scientia Sinica Mathematica, vol. 43, no. 6, pp. 551–562, 2013. [Online].Available: http://dx.doi.org/10.1360/012013-128[20] T.-H. Zhao, Y.-M. Chu, and B.-Y. Liu, “Optimal bounds for Neuman-Sa´ndor mean in terms of theconvex combinations of harmonic, geometric, quadratic, and contraharmonic means,” Abstr. Appl.Anal., pp. Art. ID 302 635, 9 pp., 2012.Authors’ addressesFeng QiDepartment of Mathematics, School of Science, Tianjin Polytechnic University, Tianjin City, 300387,China, College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mon-golia Autonomous Region, 028043, China, Institute of Mathematics, Henan Polytechnic University,Jiaozuo City, Henan Province, 454010, ChinaE-mail address: qifeng618@gmail.com, qifeng618@gmail.com, qifeng618@hotmail.com,qifeng618@hotmail.com, qifeng618@qq.com, qifeng618@qq.comURL: qifeng618.wordpress.comWen-Hui LiDepartment of Mathematics, School of Science, Tianjin Polytechnic University, Tianjin City, 300387,ChinaE-mail address: wen.hui.li102@gmail.com, wen.hui.li102@gmail.com, wen.hui.li@foxmail.com,wen.hui.li@foxmail.com

A unified proof of several inequalities and some new inequalities involving Neuman-Sandor mean

Abstract

Similar works

Full text

Available Versions

Repository of the Academy's Library