For a ? (0,1/2] and r ? (0,1), let Ka(r) (K (r)) denote the generalized elliptic integral (complete elliptic integral, respectively) of the first kind. In this article, we mainly present a sufficient and necessary condition under which the function a ? [K(r)-Ka(r)]=(1-2a)?(?? R) is monotone on (0,1/2) for each fixed r ? (0,1). The obtained result leads to the conclusion that inequality K (r)- (1-2a)? [K(r)- ?/2] ? Ka(r) ? K (r)-(1-2a)? [K(r)-?/2] holds for all a ? (0,1/2] and r ? (0,1) with the best possible constants ? = ?/2 and ? = 2.

This article defines two common $q$-orthogonal polynomials: homogeneous $q$-Laguerre polynomials and homogeneous little $q$-Jacobi polynomials. They can be viewed separately as solutions to two $q$-partial differential equations. Furthermore, an analytic function satisfies a certain system of $q$-partial differential equations if and only if it can be expanded in terms of homogeneous $q$-Laguerre polynomials or homogeneous little $q$-Jacobi polynomials. As applications, several generalized Ramanujan $q$-beta integrals and Andrews-Askey integrals are obtained.

Motivated by Liu's recent work in \cite{Liu2022}. We shall reveal the essential feature of Hahn polynomials by presenting two new $q$-exponential operators. These lead us to use a systematic method to study identities involving Hahn polynomials. As applications, we use the method of $q$-exponential operator to prove the bilinear generating function of Hahn polynomials and Heine's second transformation formula. Moreover, a generalization of $q$-Gaussian summation is given, too.