Abstract
The paper studies logarithmic convexity and concavity of the generalized hypergeometric function with respect to the simultaneous shift of several parameters. We use integral representations and properties of Meijer’s G function to prove log-convexity. When all parameters are shifted we use series manipulations to examine the power series coefficients of the generalized Turánian formed by the generalized hypergeometric function. In cases when all zeros of the generalized hypergeometric function are real, we further explore the consequences of the extended Laguerre inequalities and formulate a conjecture about reality of zeros.
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The first author was supported by NSFC (grant 11650110426).
The second author was supported by the Ministry of Education of the Russian Federation (project 1398.2014) and by the Russian Foundation for Basic Research (project 15-56-53032).
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Kalmykov, S.I., Karp, D.B. Log-concavity and Turán-type inequalities for the generalized hypergeometric function. Anal Math 43, 567–580 (2017). https://doi.org/10.1007/s10476-017-0503-z
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DOI: https://doi.org/10.1007/s10476-017-0503-z
Keywords
- generalized hypergeometric function
- Meijer’s G function
- integral representation
- log-convexity
- log-concavity
- generalized Turánian
- extended Laguerre inequality