In this paper, we find the greatest value p = log 2/（log Tr - log 2） = 1.53.- and the least value q -- 5/3 - 1.66.. such that the double inequality Mp（a,b） 〈 T（a,b） 〈 Mq（a,b） holds for all a, b 〉 0 with a # b. Here, Mp（a, b） and T（a, b） are the p-th power and Seiffertmeans of two positive numbers a and b, respectively.
we study the monotonicity of certain combinations of the Gaussian hypergeometric functions F(-1/2,1/2;1;1- xc) and F(-1/2- δ,1/2 + δ;1;1- xd) on(0,1) for given 0 < c 5d/6 < ∞ andδ∈(-1/2,1/2),and find the largest value δ1 = δ1(c,d) such that inequality F(-1/2,1/2;1;1- xc)
We study the quotient of hypergeometric functions in the theory of Ramanujan＇s generalized modular equation for a ∈ （0, 1/2], and find an infinite product for- mula for μ1/3（r） by use of the properties of μ＊a（r） and Ramanujan＇s cubic transformation. Besides, a new cubic transformation formula of hypergeometric function is given, which complements the Ramanujan＇s cubic transformation.
