Monotonicity of three functions involving the confluent hypergeometric functions of the second kind

Abstract In this paper, we approach the monotonicity of three functions involving the confluent hypergeometric function of the second kind from the perspective of using the monotonicity rules. By employing the monotonicity of these functions, we establish bounds for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>U</mml:mi> <mml:mi>′</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>/</mml:mo> <mml:mi>U</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> and U ( a , b , y )/ U ( a , b , x ), which are shown to be extremely tight for large values of x and y . Furthermore, by using the relationships between the confluent hypergeometric function of the second kind and other special functions, we derive a series of results concerning the incomplete gamma function and the modified Bessel function of the second kind, including the bounds for Γ( γ , x ), K ν +1 ( x )/ K ν ( x ), and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>K</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>ν</mml:mi> </mml:mrow> </mml:msub> <mml:mi>′</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>/</mml:mo> <mml:msub> <mml:mrow> <mml:mi>K</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>ν</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> .