In this paper, we obtain extensions of majorization type results and extensions of weighted Favard’s and Berwald’s inequality. We prove positive semi-definiteness of matrices generated by differences deduced from majorization type results and differences deduced from weighted Favard’s and Berwald’s inequality. This implies a surprising property of exponentially convexity and log-convexity of th...

We prove a conjecture of Liu and Wang on the q-log-convexity of the polynomial sequence { ∑n k=0 ( n k )2 q}n≥0. By using Pieri’s rule and the Jacobi-Trudi identity for Schur functions, we obtain an expansion of a sum of products of elementary symmetric functions in terms of Schur functions with nonnegative coefficients. Then the principal specialization leads to the q-log-convexity. We also pr...

The monotonicity, convexity, log-convexity and complete monotonicity properties for some special functions are proved, and some inequalities are established.

We find sufficient conditions for log-convexity and log-concavity for the functions of the forms a 7→ ∑ fk(a)kx , a 7→ ∑ fkΓ(a + k)x k and a 7→ ∑ fkx k/(a)k. The most useful examples of such functions are generalized hypergeometric functions. In particular, we generalize the Turán inequality for the confluent hypergeometric function recently proved by Barnard, Gordy and Richards and log-convexi...