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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.

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.

In this paper, we prove that the double inequality \begin{equation*} 1+\alpha r'^2, Comment: 14 pages

In this paper, we present several bounds for the perimeter of an ellipse in terms of arithmetic, geometric, and harmonic means, which improve some known results. Mathematics subject classification (2010): 41A10, 33E05, 33C05, 26E60.

In this paper, we present some monotonicity theorems involving the generalized elliptic integrals Ka(r) and Ea(r) , and find an asymptotic property of Ka(r) when r → 1 . As applications, some well known results are improved. Mathematics subject classification (2010): 33E05.

In this paper, we establish a necessary and sufficient condition for the concavity of the complete elliptic integrals of the second kind with respect to Holder means.

In this paper, we present several bounds for the perimeter of an ellipse, which improve some well-known results.

In this paper, we present some sharp bounds for complete elliptic integrals of the second kind. As applications, we establish several inequalities for the perimeter of an ellipse.

Genome-wide association studies have identified susceptibility loci for esophageal squamous cell carcinoma (ESCC). We conducted a meta-analysis of all single-nucleotide polymorphisms (SNPs) that showed nominally significant P-values in two previously published genome-wide scans that included a total of 2961 ESCC cases and 3400 controls. The meta-analysis revealed five SNPs at 2q33 with P5 × 10(-8), and the strongest signal was rs13016963, with a combined odds ratio (95% confidence interval) of 1.29 (1.19-1.40) and P= 7.63 × 10(-10). An imputation analysis of 4304 SNPs at 2q33 suggested a single association signal, and the strongest imputed SNP associations were similar to those from the genotyped SNPs. We conducted an ancestral recombination graph analysis with 53 SNPs to identify one or more haplotypes that harbor the variants directly responsible for the detected association signal. This showed that the five SNPs exist in a single haplotype along with 45 imputed SNPs in strong linkage disequilibrium, and the strongest candidate was rs10201587, one of the genotyped SNPs. Our meta-analysis found genome-wide significant SNPs at 2q33 that map to the CASP8/ALS2CR12/TRAK2 gene region. Variants in CASP8 have been extensively studied across a spectrum of cancers with mixed results. The locus we identified appears to be distinct from the widely studied rs3834129 and rs1045485 SNPs in CASP8. Future studies of esophageal and other cancers should focus on comprehensive sequencing of this 2q33 locus and functional analysis of rs13016963 and rs10201587 and other strongly correlated variants.

We find the greatest value α1 and α2, and the least values β1 and β2, such that the double inequalities α1S(a,b) + (1 − α1) A(a,b) 0 with a ≠ b. As applications, we get two new bounds for the complete elliptic integral of the second kind in terms of elementary functions. Here, S(a,b) = [(a2 + b2)/2]1/2, A(a,b) = (a + b)/2, and \(T(a,b)=\frac{2}{\pi}\int\limits_{0}^{{\pi}/{2}}\sqrt{a^2{\cos^2{\theta}}+b^2{\sin^2{\theta}}}{\rm d}\theta\) denote the root-square, arithmetic, and Toader means of two positive numbers a and b, respectively.

In this paper, the authors prove some inequalities for the generalized elliptic integrals with respect to Holder means.

In this paper, we establish some functional inequalities for the generalized HerschPfluger distortion function φK(a,r) , and prove a submultiplicative property for the generalized Agard distortion function ηK(a,x) . Mathematics subject classification (2010): 33E05, 30C62.

In this paper, we study the monotonicity properties of the generalized elliptic integrals and establish two Holder mean inequalities for the generalized Grotzsch ring function μa(r) and the generalized Hersch-Pfluger distortion function φa K(r) . Mathematics subject classification (2010): 30C62, 33E05.

In this paper, the authors show a relation between the generalized convexity and super- (sub-)multiplicative property, and discuss some generalized convexity and inequalities involving the Gaussian hypergeometric function, the generalized η-distortion function η K a ( x ) and the generalized Grotzsch function μ a ( r ) .

In this paper, some monotoneity and concavity properties of the gamma, beta and psi functions are obtained, from which several asymptotically sharp inequalities follow. Applying these properties, the authors improve some well-known results for the volume Ω n \Omega _n of the unit ball B n ⊂ R n B^n\subset \mathbb {R}^n , the surface area ω n − 1 \omega _{n-1} of the unit sphere S n − 1 S^{n-1} , and some related constants.

For and , we prove that the elliptic Neuman mean is strictly Schur quadratically concave on if and only if . As an application, the bounds for elliptic Neuman mean in terms of the quadratic mean are presented. MSC:26B25, 26E60.

Forp∈[0,1], the generalized Seiffert mean of two positive numbersaandbis defined bySp(a,b)=p(a-b)/arctan[2p(a-b)/(a+b)], 00witha≠b, and give new bounds for the complete elliptic integrals of the second kind. Here,T(a,b)=(2/π)∫0π/2a2cos⁡2θ+b2sin⁡2θdθdenotes the Toader mean of two positive numbersaandb.

Nat. Genet. 42, 759–763 (2010); published online 22 August 2010; corrected after print 27 August 2014 In contrast to the version of this article initially published, the authors now find no evidence to support association with esophageal squamous cell carcinoma susceptibility for rs13042395[T] at 20p13 in their original data, in two independent sets of cases and controls collected in other Chinese populations or in the joint analysis of these three studies.

We prove monotonicity properties of certain combinations of complete elliptic integrals of the first and second kind, K and E. These results lead to sharp symmetrical bounds for K and E, which improve recently discovered inequalities.

In this paper the authors study a monotonicity of several functions involving m a ( r ) and μ a ( r ) . By using these results, the authors obtain some new bounds of the solutions of Ramanujan's generalized modular equations.

The authors prove a conjecture on elliptic integrals and obtain sharp bounds for φ K ( r ) and λ ( K ) . By using Teichmuller's module theorem, the authors obtain a distortion theorem of K-quasiconformal mappings on the plane.

Continuum hypothesis on nanoscales is invalid, and a differential–difference model is considered as an alternative approach to describing discontinued problems. This paper applies the homotopy perturbation method to a nonlinear differential–difference equation arising in nanotechnology. Comparison of the approximate solution with the exact one reveals that the method is very effective.

In this work, we prove that M p ( K ( r ) , E ( r ) ) > π / 2 for all r ∈ ( 0 , 1 ) if and only if p ≥ − 1 / 2 , where M p ( x , y ) denotes the power mean of order p of two positive numbers x and y , and K ( r ) and E ( r ) denote the complete elliptic integrals of the first and second kinds, respectively.

The normal swine oesophagus was heated using an intracavity microwave radiator for 30 min at 45, 46, 47, 48 and 49 degrees C. The animals were sacrificed 48 h or 30 days later. The degree of heat injury was evaluated on an histological score system with a maximum possible damage score of 100 per cent. The acute heat injury (48 h after heating) at these temperatures was 2, 17, 40, 50 and 67 per cent respectively. The main feature of acute heat damage was degenerative necrosis or thrombosis of blood vessels at the oesophageal wall. This occurred only in one out of nine cases at 45 and 46 degrees C heating, but in seven out of nine cases at 48 and 49 degrees C. The main feature of late heat damage was fibrosis. The score of chronic damage for the above groups was 12, 23, 30, 34 and 51 per cent, respectively. When animals were preheated at 44 degrees C for 30 min and then reheated at 45-49 degrees C 4 h later, we found that preheating could provide about 2 degrees C protection. From the above data, we suggest that 45 degrees C and 30 min heating is a safe dose for normal oesophageal tissue.