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Three families of two-parameter means constructed by trigonometric functions. (English) Zbl 1297.26071
Summary: In this paper, we establish three families of trigonometric functions with two parameters and prove their monotonicity and bivariate log-convexity. Based on them, three two-parameter families of means involving trigonometric functions, which include Schwab-Borchardt mean, the first and second Seiffert means, Sándor’s mean and many other new means, are defined. Their properties are given and some new inequalities for these means are proved. Lastly, two families of two-parameter hyperbolic means, which similarly contain many new means, are also presented without proofs.

##### MSC:
 26E60 Means 26D05 Inequalities for trigonometric functions and polynomials 33B10 Exponential and trigonometric functions 26A48 Monotonic functions, generalizations
##### Keywords:
trigonometric function; hyperbolic function; mean; inequality
Full Text:
##### References:
 [1] Stolarsky, KB, Generalizations of the logarithmic Mean, Math. Mag, 48, 87-92, (1975) · Zbl 0302.26003 [2] Gini, C, Di una formula comprensiva delle medie, Metron, 13, 3-22, (1938) · Zbl 0018.41404 [3] Brenner, JL; Carlson, BC, Homogeneous Mean values: weights and asymptotes, J. Math. Anal. Appl, 123, 265-280, (1987) · Zbl 0614.26002 [4] Carlson, BC, Algorithms involving arithmetic and geometric means, Am. Math. Mon, 78, 496-505, (1971) · Zbl 0218.65035 [5] Neuman, E; Sándor, J, On the Schwab-Borchardt Mean, Math. Pannon, 14, 253-266, (2003) · Zbl 1053.26015 [6] Toader, G, Some mean values related to the arithmetic-geometric Mean, J. Math. Anal. Appl, 218, 358-368, (1998) · Zbl 0892.26015 [7] Sándor, J, Toader, G: On some exponential means. Seminar on Mathematical Analysis (Cluj-Napoca, 1989-1990), Preprint, 90, “Babes-Bolyai” Univ., Cluj, 35-40 (1990) · Zbl 0752.26010 [8] Sándor, J; Toader, G, On some exponential means. part II, No. 2006, (2006) · Zbl 1153.26317 [9] Seiffert, HJ, Werte zwischen dem geometrischen und dem arithmetischen mittel zweier zahlen, Elem. Math, 42, 105-107, (1987) · Zbl 0721.26009 [10] Seiffert, HJ, Aufgabe 16, Die Wurzel, 29, 221-222, (1995) [11] Sándor, J: Trigonometric and hyperbolic inequalities. Available online at arXiv:1105.0859 (2011). e-printatarXiv.org · Zbl 0018.41404 [12] Sándor, J, Two sharp inequalities for trigonometric and hyperbolic functions, Math. Inequal. Appl, 15, 409-413, (2012) · Zbl 1242.26016 [13] Leach, EB; Sholander, MC, Extended Mean values, Am. Math. Mon, 85, 84-90, (1978) · Zbl 0379.26012 [14] Leach, EB; Sholander, MC, Extended Mean values II, J. Math. Anal. Appl, 92, 207-223, (1983) · Zbl 0517.26007 [15] Páles, Z, Inequalities for sums of powers, J. Math. Anal. Appl, 131, 265-270, (1988) · Zbl 0649.26015 [16] Páles, Z, Inequalities for differences of powers, J. Math. Anal. Appl, 131, 271-281, (1988) · Zbl 0649.26014 [17] Qi, F, Logarithmic convexities of the extended Mean values, Proc. Am. Math. Soc, 130, 1787-1796, (2002) · Zbl 0993.26012 [18] Yang Z-H: On the homogeneous functions with two parameters and its monotonicity. J. Inequal. Pure Appl. Math. 2005., 6(4): Article ID 101. Available online at http://www.emis.de/journals/JIPAM/images/155_05_JIPAM/155_05.pdf [19] Yang, Z-H, On the log-convexity of two-parameter homogeneous functions, Math. Inequal. Appl, 10, 499-516, (2007) · Zbl 1129.26009 [20] Yang, Z-H, On the monotonicity and log-convexity of a four-parameter homogeneous Mean, No. 2008, (2008) [21] Yang, Z-H, The log-convexity of another class of one-parameter means and its applications, Bull. Korean Math. Soc, 49, 33-47, (2012) · Zbl 1238.26031 [22] Yang, Z-H, New sharp bounds for logarithmic mean and identric Mean, (2013) · Zbl 1285.26054 [23] Zhu, L, Generalized lazarevićs inequality and its applications - part II, No. 2009, (2009) [24] Zhu, L, New inequalities for hyperbolic functions and their applications, (2012) [25] Bullen PS: Handbook of Means and Their Inequalities. Kluwer Academic, Dordrecht; 2003. · Zbl 1035.26024 [26] Merkle, M, Conditions for convexity of a derivative and some applications to the gamma function, Aequ. Math, 55, 273-280, (1998) · Zbl 0922.26005 [27] Mitrinović DS: Analytic Inequalities. Springer, Berlin; 1970. · Zbl 0199.38101 [28] Mitrinović DS: Elementary Inequalities. Noordhoff, Groningen; 1964. · Zbl 0121.05302 [29] Group of compilation: Handbook of Mathematics. Peoples’ Education Press, Beijing; 1979. (Chinese) [30] Yang, Z-H, New sharp bounds for identric mean in terms of logarithmic mean and arithmetic Mean, J. Math. Inequal, 6, 533-543, (2012) · Zbl 1257.26032 [31] Neuman, E, Inequalities for the Schwab-Borchardt Mean and their applications, J. Math. Inequal, 5, 601-609, (2011) · Zbl 1252.26006 [32] Neuman, E; Sándor, J, On certain means of two arguments and their extensions, Int. J. Math. Math. Sci, 16, 981-993, (2003) · Zbl 1040.26015 [33] Neuman, E; Sándor, J, On the Schwab-Borchardt Mean, Math. Pannon, 17, 49-59, (2006) · Zbl 1100.26011 [34] Witkowski, A, Interpolations of Schwab-Borchardt Mean, Math. Inequal. Appl, 16, 193-206, (2012) · Zbl 1261.26027 [35] Jagers, AA, Solution of problem 887, Nieuw Arch. Wiskd, 12, 30-31, (1994) [36] Sándor, J, On certain inequalities for means III, Arch. Math, 76, 34-40, (2001) · Zbl 0976.26015 [37] Hästö PA: A monotonicity property of ratios of symmetric homogeneous means. J. Inequal. Pure Appl. Math. 2002., 3(5): Article ID 71 [38] Sándor, J; Neuman, E, On certain means of two arguments and their extensions, Int. J. Math. Math. Sci, 2003, 981-993, (2003) · Zbl 1040.26015 [39] Chu, Y-M; Qiu, Y-F; Wang, M-K, Sharp power Mean bounds for the combination of Seiffert and geometric means, No. 2010, (2010) · Zbl 1197.26054 [40] He D, Shen Z-J: Advances in research on Seiffert mean. Commun. Inequal. Res. 2010., 17(4): Article ID 26. Available online at http://old.irgoc.org/Article/UploadFiles/201010/20101026104515652.pdf [41] Wang, S-S; Chu, Y-M, The best bounds of the combination of arithmetic and harmonic means for the seiffert’s Mean, Int. J. Math. Anal, 4, 1079-1084, (2010) · Zbl 1207.26033 [42] Wang, M-K; Qiu, Y-F; Chu, Y-M, Sharp bounds for Seiffert means in terms of Lehmer means, J. Math. Inequal, 4, 581-586, (2010) · Zbl 1204.26053 [43] Chu, Y-M; Qiu, Y-F; Wang, M-K; Wang, G-D, The optimal convex combination bounds of arithmetic and harmonic means for the seiffert’s Mean, No. 2010, (2010) · Zbl 1209.26018 [44] Liu, H; Meng, X-J, The optimal convex combination bounds for seiffert’s Mean, No. 2011, (2011) · Zbl 1221.26037 [45] Chu, Y-M; Wang, M-K; Gong, W-M, Two sharp double inequalities for Seiffert Mean, No. 2011, (2011) · Zbl 1275.26052 [46] Chu, Y-M; Hou, S-W, Sharp bounds for Seiffert mean in terms of contraharmonic Mean, No. 2012, (2012) · Zbl 1231.26034 [47] Chu, Y-M; Long, B-Y; Gong, W-M; Song, Y-Q, Sharp bounds for Seiffert and neuman-Sándor means in terms of generalized logarithmic means, No. 2013, (2013) · Zbl 1282.26046 [48] Jiang, W-D; Qi, F, Some sharp inequalities involving Seiffert and other means and their concise proofs, Math. Inequal. Appl, 15, 1007-1017, (2012) · Zbl 1253.26052 [49] Hästö, PA, Optimal inequalities between seiffert’s mean and power Mean, Math. Inequal. Appl, 7, 47-53, (2004) · Zbl 1049.26006 [50] Costin, I; Toader, G, A Nice separation of some Seiffert type means by power means, No. 2012, (2012) · Zbl 1248.26028 [51] Yang, Z-H: Sharp bounds for the second Seiffert mean in terms of power means. Available online at arXiv:1206.5494v1 (2012). e-printatarXiv.org · Zbl 1238.26031 [52] Yang, Z-H: The monotonicity results and sharp inequalities for some power-type means of two arguments. Available online at arXiv:1210.6478 (2012). e-printatarXiv.org [53] Yang, Z-H: Sharp bounds for Seiffert mean in terms of weighted power means of arithmetic mean and geometric mean. Math. Inequal. Appl. (2013, in print) · Zbl 0018.41404 [54] Chu, Y-M; Wang, M-K; Qiu, Y-F, An optimal double inequality between power-type Heron and Seiffert means, No. 2010, (2010) · Zbl 1210.26021 [55] Costin, I; Toader, G, Optimal evaluations of some Seiffert-type means by power means, Appl. Math. Comput, 219, 4745-4754, (2013) · Zbl 06447280 [56] 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, 13, 715-723, (2010) · Zbl 1204.26023 [57] Lv, Y-P; Wang, G-D; Chu, Y-M, A note on Jordan type inequalities for hyperbolic functions, Appl. Math. Lett, 25, 505-508, (2012) · Zbl 1247.26026 [58] Yang, Z-H: Refinements of Mitrinovic-Cusa inequality. Available online at arXiv:1206.4911 (2012). e-printatarXiv.org [59] Yang, Z-H, Refinements of a two-sided inequality for trigonometric functions, J. Math. Inequal, 7, 601-615, (2013) · Zbl 1298.26045
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