Three families of two-parameter means constructed by trigonometric functions

Journal of inequalities and applications

Zhen-Hang Yang¹ 

关键词: trigonometric function;    hyperbolic function;    mean;    inequality;   

DOI  :  10.1186/1029-242X-2013-541

学科分类：数学（综合）

来源: SpringerOpen

【 摘 要 】

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, 26D05, 33B10, 26A48.

【 授权许可】

CC BY