The aim of this article is to give some improvements of Jordan-Stečkin and Becker-Stark inequalities discussed in [1]. MSC: 26D15, 41A10, 42A16

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In this paper, we obtain some improved exponential approximation inequalities for the functions and , and we prove them by using the properties of Bernoulli numbers and new criteria for the monotonicity of quotient of two power series.

In this paper we present some new upper bounds of the Cusa-Huygens and the Huygens approximations. Bounds are obtained in the forms of some polynomial and some rational functions.