Monotonicity Properties And Bounds Involving The Two-parameter Generalized Gr(?)tzsch Ring Function

Special functions are widely used in mathematics,physics,engineering technology and other fields.As one of the most important special functions,Gauss hypergeometric function ₂F₁(a,b;c;x)is the solution of all second-order linear ordinary differential equations with three regular singularities,and it is closely related to geometry and other branches of mathematics.The complete elliptic integrals ?(r)and ?(r)are special cases of Gauss hypergeometric functions,the Gr(?)tzsch ring function ?(r)and Hübner upper bound function are both defined by complete elliptic integrals,these functions play an important role in the study of classical Ramanujan modular equations.Therefore,the further exploration of them is beneficial to promote the development of special function theory,geometric function theory,quasi-conformal theory and generalized Ramanujan modular equation.In this paper,we define a class of two-paramater generalized elliptic integrals ?_a,b(r),?_a,b(r),generalized Grotzsch ring function ?_a,b(r)and m_a,b(r)in the theory of Ramanujan's generalized modular equation when c=(a+b+1)/2.The main highlight of this paper is to extend the Ramanujan's asymptotic formula to the case of non-zero balanced,and get the explicit asymptotic expansion of ?_a,b(r)near r=1.By using the monotonicity lemma of the ratio of two series,we study the monotonicity properties of rational combinations of ?_a,b(r),?_a,b(r),?_a,b(r)and m_a,b(r)with their approximation functions,and obtain the related double inequalities.There are five chapters as follows:Chapter 1:In this chapter,we mainly elaborate the research significance and history of this topic,and introduce the development and recent research of some special functions.In the meantime,we introduce the innovations of the results in this paper,and some concepts and notations involved in the subsequent chapters.Chapter 2:The definitions and properties of Gauss hypergeometric function and Grotzsch ring function are presented in this chapter,as well as two important lemmas are led to prove the monotonicity properties of functions,which are prepared for our main research conclusions.Chapter 3:In the case of non-zero balanced,that is,a+b>1,we discuss the monotonicity properties of some rational combinatorial functions about ?_a,b(r),?_a,b(r),?_a,b(r)and m_a,b(r).To this end,we put forward a conjecture according to the existing conclusion.Chapter 4:We prove the monotonicity of a function involving ?_a,b(r)and ?_a,b(r)under certain conditions of parameters,which establish an inequality,and also is a partial answer to the conjecture raised in the end of Chapter 3.Chapter 5:In this chapter,we point out the problems that need to be solved in the future,provide the ideas for enlarging the parameters'region,and propose the prospects and research directions based on the conclusions obtained so far.