New Bounds for Arithmetic Mean by the Seiffert-like Means

Ling Zhu
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Ling Zhu: Department of Mathematics, Zhejiang Gongshang University, Hangzhou 310018, China

Mathematics, 2022, vol. 10, issue 11, 1-14

Abstract: By using the power series of the functions 1 / sin n t and cos t / sin n t ( n = 1 , 2 , 3 , 4 , 5 ), and the estimation of the ratio of two adjacent Bernoulli numbers, we obtained new bounds for arithmetic mean A by the weighted arithmetic means of M tan 1 / 3 M sin 2 / 3 and 1 3 M tan + 2 3 M sin , M tanh 1 / 3 M sinh 2 / 3 and 1 3 M tanh + 2 3 M sinh , where M tan ( x , y ) and M sin ( x , y ) , M tanh ( x , y ) and M sinh ( x , y ) are the tangent mean, sine mean, hyperbolic tangent mean and hyperbolic sine mean, respectively. The upper and lower bounds obtained in this paper are compared in detail with the conclusions of the previous literature.

Keywords: bounds; arithmetic mean; Seiffert-like means; tangent mean; hyperbolic sine mean; sine mean; hyperbolic tangent mean; the ratio of two adjacent Bernoulli numbers (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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