A note on Turán type and mean inequalities for the Kummer function

Barnard, Roger W.; Gordy, Michael B.; Richards, Kendall C.

doi:10.1016/j.jmaa.2008.08.024

Cited by 38 publications

(36 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…We also extend some results of Baricz and provide sufficient conditions for log-concavity and log-convexity of the generalized hypergeometric functions which are less restrictive then the conditions a i > b i for the (less general) Turán type inequality given in [5]. We use the generalized Stieltjes transform representation for q+1 F q from [15] to extend this results to negative x.…”

Section: Introductionmentioning

confidence: 71%

“…, where f is given by (5). Now the estimate from above is just Theorem 1 while the estimate from below is Theorem 2.…”

Section: Proof Again By Direct Multiplication We Havementioning

confidence: 94%

“…Using a clever combination of contiguous relations and telescoping sums Barnard, Gordy and Richards have recently shown in [5] that this is indeed true and even more general inequality…”

Section: Introductionmentioning

confidence: 94%

“…Following the idea from [5] we apply the Kummer transformation 1 F 1 (a; c; x) = e x 1 F 1 (c − a; c; −x) so that the left-hand side of the above inequality equals e 2x…”

Section: Applications To Hypergeometric Functionsmentioning

confidence: 99%

“…Namely, it has been demonstrated in [19] that they are dependent on certain monotonicity properties of the coefficients of three-term recurrence relations. See further development in [6].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Log-convexity and log-concavity of hypergeometric-like functions

Karp¹,

Sitnik²

2010

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

We find sufficient conditions for log-convexity and log-concavity for the functions of the forms $a\mapsto\sum{f_k}(a)_kx^k$, $a\mapsto\sum{f_k}\Gamma(a+k)x^k$ and $a\mapsto\sum{f_k}x^k/(a)_k$. The most useful examples of such functions are generalized hypergeometric functions. In particular, we generalize the Tur\'{a}n inequality for the confluent hypergeometric function recently proved by Barnard, Gordy and Richards and log-convexity results for the same function recently proved by Baricz. Besides, we establish a reverse inequality which complements naturally the inequality of Barnard, Gordy and Richards. Similar results are established for the Gauss and the generalized hypergeometric functions. A conjecture about monotonicity of a quotient of products of confluent hypergeometric functions is made.Comment: 13 pages, no figure

show abstract

Section: Introductionmentioning

confidence: 71%

“…, where f is given by (5). Now the estimate from above is just Theorem 1 while the estimate from below is Theorem 2.…”

Section: Proof Again By Direct Multiplication We Havementioning

confidence: 94%

“…Using a clever combination of contiguous relations and telescoping sums Barnard, Gordy and Richards have recently shown in [5] that this is indeed true and even more general inequality…”

Section: Introductionmentioning

confidence: 94%

“…Following the idea from [5] we apply the Kummer transformation 1 F 1 (a; c; x) = e x 1 F 1 (c − a; c; −x) so that the left-hand side of the above inequality equals e 2x…”

Section: Applications To Hypergeometric Functionsmentioning

confidence: 99%

“…Namely, it has been demonstrated in [19] that they are dependent on certain monotonicity properties of the coefficients of three-term recurrence relations. See further development in [6].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations