Journal of Inequalities and Applications is a peer-reviewed open access journal published under the brand SpringerOpen. The aim of this journal is to provide ...

The authors establish a pair of closed-form expressions for special values of the Bell polynomials of the second kind for the falling factorials, derive two pairs of identities involving the falling factorials, find an equivalent expression between two special values for the Bell polynomials of the second kind, and present five closed-form expressions for the (modified) spherical Bessel functions.

In the article, a general double refinement of Jordan's inequality, for n∈ℕ, is established, where the coefficients α _k and β _k defined by recursing formulas (9) and (10) are the best possible. As an application, L. Yang's inequality is refined.

Viewing from polynomials solutions of q-difference equations makes certain generating functions calculated easily. In this short paper, we build the relations between Cigler’s polynomials and q-difference equations. In addition, we deduce generating functions and bilinear generating functions for Cigler’s polynomials. More over, we gain multilinear and mixed generating functions for Cigler’s polynomials. At last, we gain $U(n+1)$ type generating functions for Cigler’s polynomials.

In the paper, the famous Hermite–Hadamard integral inequality for convex functions is generalized to and refined as the ones for n -time differentiable functions which are s -convex in the second sense, and some known results are improved.