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The authors present sharp transformation inequalities for the zero-balanced hypergeometric function $${}_2F_1(a,b;a+b;r)$$ created by the transformations $$r\mapsto x=[(1-r)/(1+2r)]^3$$ and $$r\mapsto 1-x$$ , $$r\mapsto u=(1/2)r(3+r)^2(1+r)^{-3}$$ and $$r\mapsto 1-u$$ , $$r\mapsto p=(27/2)r(1+r)^4(1+4r+r^2)^{-3}$$ and $$r\mapsto 1-p$$ , by showing the monotonicity properties of certain combinations in terms of hypergeometric functions and elementary functions, thus extending the transformation identities satisfied by $${}_2F_1(1/3,2/3;1;r)$$ with these three pairs of transformations and substantively improving the related known results. With these results, some properties are obtained for the generalized Grotzsch ring functions and the modular functions appearing in Ramanujan’s modular equations. Some other properties of $${}_2F_1(a,b;c;r)$$ are obtained, too.

The authors present sharp Landen transformation inequalities for the hypergeometric functions F 1 2 ( a , b ; a + b ; x ) and F 1 2 ( a , b ; ( a + b + 1 ) / 2 ; x ) , by showing the monotonicity properties of certain combinations defined in terms of one of these two hypergeometric functions and linear (or rational) functions, thus giving complete solutions of the problem on extending the well-known Landen transformation identities for the complete elliptic integrals of the first kind to these two hypergeometric functions, and substantially improving the related known results. As applications of these results, sharp Landen transformation inequalities are obtained for the generalized Grotzsch ring functions and the modular functions, which appear in Ramanujan's modular equations. Some other properties of hypergeometric functions and several properties of the Ramanujan constant are obtained, too.

In this paper, the authors present sharp approximations in terms of sine function and polynomials for the so-called Ramanujan constant (or the Ramanujan $R$-function) $R(a)$, by showing some monotonicity, concavity and convexity properties of certain combinations defined in terms of $R(a)$, $\sin(\pi a)$ and polynomials. Some properties of the Riemann zeta function and its related special sums are presented, too., Comment: 19 pages

The authors obtain some monotonicity and concavity properties, and asymptotically sharp bounds for the generalized elliptic integral K a ( r ) of the first kind and its special case K ( r ) , the complete elliptic integral of the first kind, by studying the monotonicity and concavity properties of certain combinations defined in terms of the Gaussian hypergeometric function F ( a , 1 − a ; 1 ; x ) and elementary functions, thus improving some well-known results.

The authors present the power series expansions of the function R ( a ) − B ( a ) at a = 0 and at a = 1 / 2 , show the monotonicity and convexity properties of certain familiar combinations defined in terms of polynomials and the difference between the so-called Ramanujan constant R ( a ) and the beta function B ( a ) ≡ B ( a , 1 − a ) , and obtain asymptotically sharp lower and upper bounds for R ( a ) in terms of B ( a ) and polynomials. In addition, some properties of the Riemann zeta function ζ ( n ) , n ∈ N , and its related sums are derived.

In this paper, the authors present sharp bounds of the Hubner function M , which is defined by (1.11) and has important applications in the theories of quasiconformal maps and Ramanujan's modular equations, by showing the monotonicity and concavity-convexity properties of certain combinations defined in terms of M and elementary functions. By these results, several well-known results for M including its bounds and logarithmic inequalities, the explicit quasiconformal Schwarz lemma and the estimates of the solutions to Ramanujan's classical modular equations are remarkably improved. A simpler and more concise proof of the series expansion of M ( r ) is given, too.

In this paper, we study the quotient of hypergeometric functions μ a (r) in the theory of Ramanujan’s generalized modular equation for a ∈ (0, 1/2]. Several new inequalities are given for this and related functions. Our main results complement and generalize some known results in the literature.

In this paper, the authors show some monotonicity and concavity of the classical psi function, by which several known results are improved and some new asymptotically sharp estimates are obtained for this function. In addition, applying the new results to the psi function, the authors improve the well-known lower and upper bounds for the approximate evaluation of Euler’s constant γ.

For a ∈ ( 0 , 1 / 2 ] and r ∈ ( 0 , 1 ) , let K a ( r ) and E a ( r ) ( K ( r ) and E ( r ) ) denote the generalized elliptic integrals (the complete elliptic integrals, respectively) of the first and second kinds, respectively. In this paper, the authors present the necessary and sufficient conditions under which certain familiar combinations defined in terms of the generalized elliptic integrals and elementary functions are monotone in a ∈ ( 0 , 1 / 2 ] . By these results, sharp bounds expressed in terms of K ( r ) , E ( r ) and elementary functions are obtained for K a ( r ) and E a ( r ) , thus showing the dependence on the parameter a of K a ( r ) and E a ( r ) , and substantially improving the known related results.