Sharp inequalities for tangent function with applications

Abstract

In the article, we present new bounds for the function on the interval and find sharp estimations for the Sine integral and the Catalan constant based on a new monotonicity criterion for the quotient of power series, which refine the Redheffer and Becker-Stark type inequalities for tangent function.

New bounds for the exponential function with cotangent

“…And, since is decreasing in p , it follows that, for , These imply that the inequality ( 5.4 ) holds. It thus can be seen that the above theorem partly refines Lv et al’s result [ 19 ].…”

“…The second aim of this paper is to refine some known results presented in [ 19 ], we shall state it carefully in the fifth section.…”

“…[ 2 – 18 ]. Very recently, Lv, Yang, Luo, and Zheng [ 19 ] gave a new type of bounds for the function . More precisely, they proved the following results.…”

Monotonicity rules for the ratio of two Laplace transforms with applications

“…The auxiliary function H f,g and its properties are very helpful to investigate those monotonicity of ratios of two functions, see [11], [12], [13], [14], [15], [16], [17], [18], [19], [20]. Recently, in [21] they were successfully applied to establish monotonicity rules for ratios of two power series and of two polynomials under the condition that the ratio of coefficients of two power series is increasing (decreasing) then decreasing (increasing).…”

Notes on a Double Inequality for Ratios of any Two Neighbouring Non-zero Bernoulli Numbers

“…is strictly increasing (or strictly decreasing, respectively) if and only if 1 θ ≤ − (or 0 θ ≥ , respectively). The double inequality (2) has been cited and applied in the papers [22][23][24][25][26][27][28][29].…”