Geodesic
On two new means of two arguments~III
Problemy analiza, Tome 7 (2018) no. 1, pp. 116-133.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper we establish two sided inequalities for the following two new means $$ X=X(a,b)=Ae^{G/P-1},\qquad Y=Y(a,b)=Ge^{L/A-1}, $$ where $A$, $G$, $L$ and $P$ are the arithmetic, geometric, logarithmic, and Seiffert means, respectively. As an application, we refine many other well known inequalities involving the identric mean $I$ and the logarithmic mean $L$.
Keywords: inequalities; means of two arguments; identric mean; logarithmic mean. 
@article{PA_2018_7_1_a7,
     author = {J. S\'andor and B. A. Bhayo},
     title = {On two new means of two {arguments~III}},
     journal = {Problemy analiza},
     pages = {116--133},
     publisher = {mathdoc},
     volume = {7},
     number = {1},
     year = {2018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/PA_2018_7_1_a7/}
}
TY  - JOUR
AU  - J. Sándor
AU  - B. A. Bhayo
TI  - On two new means of two arguments~III
JO  - Problemy analiza
PY  - 2018
SP  - 116
EP  - 133
VL  - 7
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/PA_2018_7_1_a7/
LA  - en
ID  - PA_2018_7_1_a7
ER  - 
%0 Journal Article
%A J. Sándor
%A B. A. Bhayo
%T On two new means of two arguments~III
%J Problemy analiza
%D 2018
%P 116-133
%V 7
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/PA_2018_7_1_a7/
%G en
%F PA_2018_7_1_a7
J. Sándor; B. A. Bhayo. On two new means of two arguments~III. Problemy analiza, Tome 7 (2018) no. 1, pp. 116-133. http://geodesic.mathdoc.fr/item/PA_2018_7_1_a7/

[1] Alzer H., “Ungleichungen fur Mittelwerte”, Arch. Math. (Basel), 47 (1986), 422–426 | DOI | MR | Zbl

[2] Alzer H., Qiu S.-L., “Inequalities for means in two variables”, Arch. Math., 80:2 (2003), 201–215 | DOI | MR | Zbl

[3] Alzer H., “Two inequalities for means”, C.R. Math. Rep. Acad. Sci. Canada, 9 (1987), 11–16 | MR | Zbl

[4] Anderson G. D., Vamanamurthy M. K., Vuorinen M., “Monotonicity Rules in Calculus”, Amer. Math. Monthly, 113:9 (2006), 805–816 | DOI | MR | Zbl

[5] Bhayo B. A., Sandor J., “On two new means of two variables II”, Notes Number Th. Discr. Math., 20:4 (2014), 1–10 | Zbl

[6] Bhayo B. A., Sandor J., “On some inequalities for the identric, logarithmic and related means”, J. Math. Ineq., 9:3 (2015), 889–896 | MR | Zbl

[7] Bhayo B. A., Sandor J., “On certain old and new trigonometric and hyperbolic inequalities”, Analysis Mathematica, 41:1–2 (2015), 3–15 | DOI | MR | Zbl

[8] Biernacki M., Krzyż J., “On the monotonicity of certain functionals in the theory of analytic functions”, Ann. Univ. Mariae Curie-Skodowska, 2 (1955), 134–145 | DOI | MR

[9] Carlson B. C., “The logarithmic mean”, Amer. Math. Monthly, 79 (1972), 615–618 | DOI | MR | Zbl

[10] Chu Y.-M., Yang Z.-H., Wu L.-M., “Sharp power mean bounds for Sandor mean”, Abstr. Appl. Anal., 2015, 172867, 5 pp. | MR

[11] Hasto P. A., “Optimal inequalities between Seiffert's mean and power means”, Math. Inequal. Appl., 1:7 (2004), 47–53 | MR | Zbl

[12] Ireland K., Rosen M., A Classical Introduction to Modern Number Theory, 2nd ed., Springer-Verlag, New York–Berlin–Heidelberg, 1990 | MR | Zbl

[13] Jeffrey A., Handbook of Mathematical Formulas and Integrals, 3rd ed., Elsevier Academic Press, 2004 | MR | Zbl

[14] Klen R., Visuri M., Vuorinen M., “On Jordan type inequalities for hyperbolic functions”, J. Ineq. Appl., 2010, 14 | DOI | MR | Zbl

[15] Li J.-L., “An identity related to Jordans inequality”, Internat. J. Math. Math. Sci., 6 (2006), 76782 | DOI | MR | Zbl

[16] Lin T. P., “The power mean and the logarithmic mean”, Amer. Math. Monthly, 81 (1974), 879–883 | DOI | MR | Zbl

[17] Mitrinovic D. S., Analytic Inequalities, Springer-Verlag, Berlin, 1970 | MR | Zbl

[18] Mitrinovic D. S, Pecaric J. E., Fink A. M., Classical and new inequalities in analysis, Kluwer Acad. Publ., 1993, 740 pp. | MR | Zbl

[19] Neuman E., Sandor J., “Optimal inequalities for hyperbolic and trigonometric functions”, Bull. Math. Anal. Appl., 3:3 (2011), 177–181 | MR | Zbl

[20] Neuman E., Sandor J., “On some inequalities involving trigonometric and hyperbolic functions with emphasis on the Cusa-Huygens, Wilker, and Huygens inequalities”, Math. Inequal. Appl., 13:4 (2010), 715–723 | MR | Zbl

[21] Neuman E., Sandor J., “On the Schwab-Borchardt mean”, Math. Pannonica, 14:2 (2003), 253–266 | MR | Zbl

[22] Neuman E., Sandor J., “On the Schwab-Borchardt mean II”, Math. Pannonica, 17:1 (2006), 49–59 | MR | Zbl

[23] Neuman E., Sandor J., “Companion inequalities for certain bivariate means”, Appl. Anal. Discr. Math., 3:1 (2009), 46–51 | DOI | MR | Zbl

[24] Neuman E., Sandor J., “On certain means of two arguments and their extensions”, Intern. J. Math. Sci., 6 (2003), 981–993 | DOI | MR

[25] Rado T., “On convex functions”, Trans. Amer. Math. Soc., 37 (1935), 266–285 | DOI | MR

[26] Ruskeepää H., Mathematica\circledR Navigator, 3rd ed., Academic Press, 2009

[27] Sandor J., “Two sharp inequalities for trigonometric and hyperbolic functions”, Math. Inequal. Appl., 15:2 (2012), 409–413 | MR | Zbl

[28] Sandor J., “Some integral inequalities”, Elem. Math., 43:6 (1988), 177–180 | MR | Zbl

[29] Sandor J., “On two new means of two variables”, Notes Number Th. Discr. Math., 20:1 (2014), 1–9 | Zbl

[30] Sandor J., “On the identric and logarithmic means”, Aequat. Math., 40 (1990), 261–270 | DOI | MR | Zbl

[31] Sandor J., “A note on certain inequalities for means”, Arch. Math. (Basel), 56 (1991), 471–473 | DOI | MR | Zbl

[32] Sandor J., “On certain identities for means”, Studia Univ. Babes-Bolyai, Math., 38, 1993, 7–14 | MR | Zbl

[33] Sandor J., “On certain inequalities for means III”, Arch. Math. (Basel), 67 (2001), 34–40 | DOI | MR

[34] Sandor J., “New refinements of two inequalities for means”, J. Math. Ineq., 2:7 (2013), 251–254 | DOI | MR | Zbl

[35] Sandor J., “A note on log-convexity of the power means”, Annales Math. Inf., 45 (2015), 107–110 | MR | Zbl

[36] Seiffert H. J., “Comment to Proble 1365”, Math. Mag., 65 (1992), 356 | DOI | MR

[37] Seiffert H. J., “Ungleichungen für einen bestimmten Mittelwert”, Nieuw Arch. Wisk. (Ser. 4), 13 (1995), 195–198 | MR | Zbl

[38] Seiffert H. J., “Problem 887”, Nieuw. Arch. Wisk., 11 (1993), 176

[39] Seiffert H. J., “Ungleichungen fur elementare Mittelwerte”, Arch. Math. (Basel), 64 (1995), 129–131 | DOI | MR | Zbl

[40] Vamanamurthy M. K., Vuorinen M., “Inequalities for means”, J. Math. Anal. Appl., 183 (1994), 155–166 | DOI | MR | Zbl

[41] Zhou S.-S., Qiam W.-M., Chu Y.-M., Zhang X.-H., “Sharp power-type Heronian mean bounds for Sándor and Yang means”, J. Inequal. Appl., 159 (2015) | DOI | MR