The aim of this paper is to give several inequalities for power series starting from a generalization of Young’s inequality for sequences of complex numbers. Then some inequalities deduced from some variants of the arithmetic-geometric mean inequality will be given. Thus by Theorem 1, Theorem 2 and Theorem 3 several refinements of Young’s inequality for functions defined by power series with real coefficients are given and by Theorem 4 a generalization of a sharp H¨older’s inequality for functions defined by power series with real coefficients is presented. Then a generalization of Young’s inequality for m pair of complex numbers in the case of the functions defined by power series is given in Remark 1, and a variant of Muirhead’s inequality for functions defined by power series with real coefficients is given in Proposition 3. There are a lot of examples related to some fundamental complex functions such as the exponential, logarithm, trigonometric and hyperbolic functions and also there are applications for some special functions such as polylogarithm, hypergeometric and Bessel functions for the first kind. Finally, we present an application related to the average information.
Ciurdariu, Loredana and Minculete, Nicuşor
"Inequalities for Power Series,"
Applied Mathematics & Information Sciences: Vol. 09:
4, Article 20.
Available at: https://digitalcommons.aaru.edu.jo/amis/vol09/iss4/20