On majorization‎, ‎Favard and Berwald inequalities

Abstract

‎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 these differences‎ ‎which allows us to deduce Lyapunov's and Dresher's inequalities for these‎ ‎differences‎, ‎which are improvements of majorization type results and‎ ‎weighted Favard's and Berwald's inequalities‎. ‎Analogous Cauchy's‎ ‎type means‎, ‎as equivalent forms of exponentially convexity and log-convexity‎, ‎are also studied‎ ‎and the monotonicity properties are proved‎.