Inequalities for zero-balanced hypergeometric functions

Anderson, G. D.; Barnard, Roger W.; Richards, Kendall C.; Vamanamurthy, M. K.; Вуоринен, Матти

doi:10.1090/s0002-9947-1995-1264800-3

Cited by 133 publications

(34 citation statements)

References 13 publications

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“…Hence it follows from (1.11), (4.15) and (4.16) that 0 < −a 2n+2 < a 2n+1 < −a 2n , so that (4.8) holds. Similarly to (4.15), we can easily obtain 3 with P 3 (1) = 65/27 = 4µ. Hence by (1.6),…”

Section: Proof Of Theorem 12mentioning

confidence: 71%

Sharp Approximations for the Ramanujan Constant

Qiu

Huang

2019

Constr Approx

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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(πa) and polynomials. Some properties of the Riemann zeta function and its related special sums are presented, too.

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Section: Proof Of Theorem 12mentioning

confidence: 71%

Sharp Approximations for the Ramanujan Constant

Qiu

Huang

2019

Constr Approx

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“…For example, following the analogy with [2], in the zero-balanced case Corollary 3 motivates considering the monotonicity of the functions…”

Section: Conjecturementioning

confidence: 99%

The Fox-Wright function near the singularity and the branch cut

Karp

Prilepkina

2020

Journal of Mathematical Analysis and Applications

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The Fox-Wright function is a further extension of the generalized hypergeometric function obtained by introducing arbitrary positive scaling factors into the arguments of the gamma functions in the summand. Its importance comes mostly from its role in fractional calculus although other interesting applications also exist. If the sums of the scaling factors in the top and bottom parameters are equal, the series defining the Fox-Wright function has a finite non-zero radius of convergence. It was demonstrated by Braaksma in 1964 that the Fox-Wright function can then be extended to a holomorphic function in the complex plane cut along a ray from the positive point on the boundary of the disk of convergence to the point at infinity. In this paper we study the behavior of the Fox-Wright function in the neighborhood of this positive singular point. Under certain restrictions we give a convergent expansion with recursively computed coefficients completely characterizing this behavior. We further compute the jump and the average value of the Fox-Wright function on the banks of the branch cut.

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“…Since lim x30 1 x log1 À x À1, and since the function g 1 r I 1 À F aY 1 À aY 1Y r 2 alog1 À r 2 is strictly increasing from 0Y 1 onto a1 À aY 1aBaY 1 À a (see [ABRVV,Theorem 1.3(1)]), it follows from (3.7) and (2.10) that 3X8 lim r30 m a r log r lim r30 4 BaY 1 À aF aY 1 À aY 1Y r H2 2 log r 2F aY 1 À aY 1Y r 2 F aY 1 À aY 1Y r 2 À 1 F aY 1 À aY 1Y r 2 log1 À r 2 Á log1 À r 2 r 2 Á r 2 log r 5…”

Section: X7mentioning

confidence: 99%

Duplication inequalities for the ratios of hypergeometric functions

Vuorinen¹

1999

Forum Mathematicum

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The authors derive the analogs of the duplication formula and logarithmic inequalities satis®ed by mr, the modulus of the plane Gro È tzsch ring B 2 n0Y r, for its generalization m a r I pF aY 1 À aY 1Y 1 À r 2 a2F aY 1 À aY r 2 sinpa for a e 0Y 1a2 and r e 0Y 1. Sharp estimates are obtained for m a r, too.1991 Mathematics Subject Classi®cation: 33C05, 30C62; 11F99, 26D15.Brought to you by |

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Inequalities for zero-balanced hypergeometric functions

Abstract: Abstract.The authors study certain monotoneity and convexity properties of the Gaussian hypergeometric function and those of the Euler gamma function.

Cited by 133 publications

References 13 publications

Sharp Approximations for the Ramanujan Constant

Sharp Approximations for the Ramanujan Constant

The Fox-Wright function near the singularity and the branch cut

Duplication inequalities for the ratios of hypergeometric functions

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