Abstract
In this paper, we establish new sharp double inequality of Becker–Stark type by using a role of the monotonicity criterion for the quotient of power series and the estimation of the ratio of two adjacent even-indexed Bernoulli numbers. The inequality results are better than those in the existing literature.
Highlights
Becker and Stark [1] (or see Kuang [2] (5.1.102, p. 398)) obtained the following twosided rational approximation for/x: Proposition 1
Zhu [4] obtained a refinement of the Becker–Stark inequalities (1) in another way as follows
In 2015, Banjac, Markragić and Malešević [11] obtained the following results about the function/x
Summary
Becker and Stark [1] (or see Kuang [2] (5.1.102, p. 398)) obtained the following twosided rational approximation for (tan x )/x: Proposition 1. Zhu [4] obtained a refinement of the Becker–Stark inequalities (1) in another way as follows. In 2015, Banjac, Markragić and Malešević [11] obtained the following results about the function (tan x )/x. Letting λ = 30π 2 / 240 − 17π 2 we can obtain the expression of the above function without x2 , and draw the following inequality conclusion by using the property for the ratio of two adjacent even-indexed Bernoulli numbers and a role of the monotonicity criterion for the quotient of power series. Holds with the best constants 240 − 17π 2 π 2 /45 and 240 − 17π 2 1024/ π 4 17π 2 − 120
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