Monotonicity of the ratio of two arbitrary gaussian hypergeometric functions

Abstract The Gaussian hypergeometric function is an important class of special functions, whose mathematical properties have long been of interest to the academic community. In this paper, by utilizing monotonicity rules, we establish the necessary and sufficient conditions for the monotonicity of the ratio of two Gaussian hypergeometric functions <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>x</m:mi> <m:mo stretchy="false">↦</m:mo> <m:mfrac> <m:mrow> <m:mi>F</m:mi> <m:mfenced open="(" close=")"> <m:mrow> <m:msub> <m:mi>a</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mi>b</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>;</m:mo> <m:msub> <m:mi>c</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>;</m:mo> <m:mi>x</m:mi> </m:mrow> </m:mfenced> </m:mrow> <m:mrow> <m:mi>F</m:mi> <m:mfenced open="(" close=")"> <m:mrow> <m:msub> <m:mi>a</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mi>b</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>;</m:mo> <m:msub> <m:mi>c</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>;</m:mo> <m:mi>x</m:mi> </m:mrow> </m:mfenced> </m:mrow> </m:mfrac> </m:math> $x \mapsto \frac{F\left(a_1, b_1 ; c_1 ; x\right)}{F\left(a_2, b_2 ; c_2 ; x\right)}$ for positive parameters a i , b i , c i ( i = 1, 2). Subsequently, we also derive some corollaries and provide some conclusions related to the complete elliptic integral function and its variants.