# zbMATH — the first resource for mathematics

Sharp one-parameter mean bounds for Yang mean. (English) Zbl 1400.26071
Summary: We prove that the double inequality $$J_\alpha(a, b) < U(a, b) < J_\beta(a, b)$$ holds for all $$a, b > 0$$ with $$a \neq b$$ if and only if $$\alpha \leq \sqrt{2} /(\pi - \sqrt{2}) = 0.8187 \cdots$$ and $$\beta \geq 3 / 2$$, where $$U(a, b) = (a - b) / [\sqrt{2} \arctan((a - b) / \sqrt{2 a b})]$$, and $$J_p(a, b) = p(a^{p + 1} - b^{p + 1}) / [(p + 1)(a^p - b^p)](p \neq 0, - 1)$$, $$J_0(a, b) = (a - b) /(\log a - \log b)$$, and $$J_{- 1}(a, b) = a b(\log a - \log b) /(a - b)$$ are the Yang and $$p$$th one-parameter means of $$a$$ and $$b$$, respectively.

##### MSC:
 26E60 Means 26D07 Inequalities involving other types of functions
Full Text:
##### References:
  Alzer, H., Über eine Einparametrige Familie von Mittelwerten, (1988), Die Bayerische Akademie der Wissenschaften · Zbl 0601.26015  Cheung, W.-S.; Qi, F., Logarithmic convexity of the one-parameter mean values, Taiwanese Journal of Mathematics, 11, 1, 231-237, (2007) · Zbl 1132.26324  Qi, F.; Cerone, P.; Dragomir, S. S.; Srivastava, H. M., Alternative proofs for monotonic and logarithmically convex properties of one-parameter mean values, Applied Mathematics and Computation, 208, 1, 129-133, (2009) · Zbl 1160.26008  Wang, M.-K.; Qiu, Y.-F.; Chu, Y.-M., An optimal double inequality among the oneparameter, arithmetic and harmonic means, Revue d’Analyse Numérique et de Théorie de l’Approximation, 39, 2, 169-175, (2010)  He, Z.-Y.; Wang, M.-K.; Chu, Y.-M., Optimal one-parameter mean bounds for the convex combination of arithmetic and logarithmic means, Journal of Mathematical Inequalities, 9, 3, 699-707, (2015) · Zbl 1333.26035  Xia, W.-F.; Hou, S.-W.; Wang, G.-D.; Chu, Y.-M., Optimal one-parameter mean bounds for the convex combination of arithmetic and geometric means, Journal of Applied Analysis, 18, 2, 197-207, (2012) · Zbl 1276.26063  Gao, H.-Y.; Niu, W.-J., Sharp inequalities related to one-parameter mean and Gini mean, Journal of Mathematical Inequalities, 6, 4, 545-555, (2012) · Zbl 1257.26028  Hu, H.-N.; Tu, G.-Y.; Chu, Y.-M., Optimal bounds for Seiffert mean in terms of one-parameter means, Journal of Applied Mathematics, 2012, (2012) · Zbl 1322.26004  Song, Y.-Q.; Xia, W.-F.; Shen, X.-H.; Chu, Y.-M., Bounds for the identric mean in terms of one-parameter mean, Applied Mathematical Sciences, 7, 88, 4375-4386, (2013)  Xia, W.-F.; Wang, G.-D.; Chu, Y.-M.; Hou, S.-W., Sharp inequalities between one-parameter and power means, Advances in Mathematics, 42, 5, 713-722, (2013) · Zbl 1299.26069  Yang, Z.-H., Three families of two-parameter means constructed by trigonometric functions, Journal of Inequalities and Applications, 2013, article 541, (2013) · Zbl 1297.26071  Yang, Z.-H.; Chu, Y.-M.; Song, Y.-Q.; Li, Y.-M., A sharp double inequality for trigonometric functions and its applications, Abstract and Applied Analysis, 2014, (2014)  Yang, Z.-H.; Wu, L.-M.; Chu, Y.-M., Optimal power mean bounds for Yang mean, Journal of Inequalities and Applications, 2014, article 401, (2014) · Zbl 1339.26086  Zhou, S.-S.; Qian, W.-M.; Chu, Y.-M.; Zhang, X.-H., Sharp power-type Heronian mean bounds for the Sándor and Yang means, Journal of Inequalities and Applications, 2015, article 159, (2015) · Zbl 1372.26032
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.