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.

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In this paper, by observing the monotonicity of three ratios involving the integral function x 0 e −t p dt for p,x > 0 , we offer some new sharp bounds for the incomplete gamma function, which greatly improve and extend some known results. Also, as by-products, we unexpectedly obtain two power series representations for the incomplete gamma function. (2010): 33B20, 26A48, 26D15. Mathematics subject classification