×

zbMATH — the first resource for mathematics

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.

MSC:
26D07 Inequalities involving other types of functions
PDF BibTeX XML Cite
Full Text: DOI
References:
[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.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.