Landen Inequalities for Special Functions

These Landen identities, which are in fact equivalent to each other, have been the starting points of the investigations of Qiu and Vuorinen [13], and recently of Simić and Vuorinen [14]. In this paper, motivated by [14], we make a contribution to the subject by showing that [14, Theorem 2.1], proved for the zero-balanced hypergeometric function F (a, b; a + b; ·), can be extended to the hypergeometric function F (a, b; c; ·) and also to general power series. Moreover, we prove that, by using a generalization of the first Landen identity in (1), the Landen inequalities for the Gaussian hypergeometric functions can be improved in some cases. Our main results complement the results from [6, 8, 13] and [14].