New Bounds for Arithmetic Mean by the Seiffert-like Means

Abstract

By using the power series of the functions 1/sinnt and cost/sinnt (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 Mtan1/3Msin2/3 and 13Mtan+23Msin,Mtanh1/3Msinh2/3 and 13Mtanh+23Msinh, where Mtan(x,y) and Msin(x,y), Mtanh(x,y) and Msinh(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