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Alternative proofs for monotonic and logarithmically convex properties of one-parameter mean values. (English) Zbl 1160.26008
Simpler proofs for results originally due to H. Alzer [Bayer. Akad. Wiss. Math.-Natur Kl. Sitzungsber. 1987, 1–9 (1988; Zbl 0601.26015) and ibid. 1988, 23–29 (1989; Zbl 0688.26009)] are given.

26D07 Inequalities involving other types of functions
Full Text: DOI
[1] Alzer, H., On stolarsky’s Mean value family, Internat. J. math. ed. sci. tech., 20, 186-189, (1987) · Zbl 0671.26009
[2] Alzer, H., Über eine einparametrige familie von mittelwerten (on a one-parameter family of means), Bayer. akad. wiss. math.-natur. kl. sitzungsber., 1987, 1-9, (1988), (in German)
[3] Alzer, H., Über eine einparametrige familie von mittelwerten. II (on a one-parameter family of Mean values. II), Bayer. akad. wiss. math.-natur. kl. sitzungsber., 1988, 23-29, (1989), (in German) · Zbl 0688.26009
[4] Bullen, P.S., A dictionary of inequalities, Pitman monographs and surveys in pure and applied mathematics, vol. 97, (1998), Addison-Wesley Longman Limited Harlow, Essex
[5] W.-S. Cheung, F. Qi, Logarithmic convexity of the one-parameter mean values, Taiwanese J. Math. 11 (2007), 231-237; RGMIA Res. Rep. Coll. 7 (2) (2004), Article 15, 331-342. Available from: <http://www.staff.vu.edu.au/rgmia/v7n2.asp>. · Zbl 1132.26324
[6] Guo, B.-N.; Liu, A.-Q.; Qi, F., Monotonicity and logarithmic convexity of three functions involving exponential function, J. Korea soc. math. ed. ser. B pure appl. math., 15, 387-392, (2008) · Zbl 1179.26035
[7] Kuang, J.-C., Chángyòng Bùděngshı‘ (applied inequalities), (2004), Shandong Science and Technology Press Ji’nan City, Shandong Province, China, (in Chinese) · Zbl 0744.26010
[8] Liu, A.-Q.; Li, G.-F.; Guo, B.-N.; Qi, F., Monotonicity and logarithmic concavity of two functions involving exponential function, Internat. J. math. ed. sci. tech., 39, 686-691, (2008)
[9] Qi, F., Logarithmic convexity of extended Mean values, Proc. amer. math. soc., 130, 1787-1796, (2002) · Zbl 0993.26012
[10] F. Qi, Logarithmic convexities of the extended mean values, RGMIA Res. Rep. Coll. 2 (5) (1999), Article 5, 643-652. Available from: <http://www.staff.vu.edu.au/rgmia/v2n5.asp>.
[11] F. Qi, The extended mean values: definition, properties, monotonicities, comparison, convexities, generalizations, and applications, Cubo Mat. Ed. 5 (3) (2003), 63-90; RGMIA Res. Rep. Coll. 5 (1) (2002), Article 5, 57-80. Available from: <http://www.staff.vu.edu.au/rgmia/v5n1.asp>.
[12] F. Qi, B.N. Guo, The function \((b^x - a^x) / x\): logarithmic convexity, RGMIA Res. Rep. Coll. 11 (1) (2008), Article 5, 1-9. Available from: <http://www.staff.vu.edu.au/rgmia/v11n1.asp>.
[13] Qi, F.; Guo, S.; Guo, B.-N.; Chen, S.-X., A class of k-log-convex functions and their applications to some special functions, Integral transform. spec. fun., 19, 195-200, (2008) · Zbl 1141.26004
[14] Qi, F.; Xu, S.-L., Refinements and extensions of an inequality. II, J. math. anal. appl., 211, 616-620, (1997) · Zbl 0932.26010
[15] Qi, F.; Xu, S.-L., The function \((b^x - a^x) / x\): inequalities and properties, Proc. amer. math. soc., 126, 3355-3359, (1998) · Zbl 0904.26006
[16] S.-Q. Zhang, B.-N. Guo, F. Qi, A concise proof for properties of three functions involving exponential function, Appl. Math. E-Notes 9 (2009), in press.
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