Functional Inequalities and Bounds for the Generalized Marcum Function of the Second Kind

Abstract

In this paper, we consider the generalized Marcum function of the second kind as an analogous function of the so-called generalized Marcum Q-function. We provide the log-convexity (log-concavity) property for its unit complement and improve some of our previous results on it. One of the transformed functions of the generalized Marcum function of the second kind is discussed in details in this paper. This form of the generalized Marcum function of the second kind supplies various important inequalities. We also discuss the Turán type inequality for the generalized Marcum Q-function. Additionally, we provide the bounds for the generalized Marcum function of the second kind as well as for its symmetric difference.

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Acknowledgements

N. Bisht remains grateful to the Council of Scientific and Industrial Research India (Grant No. 09/1022(0014)/2013-EMR-I) for financial support. S. Singh would like to thank the Science and Engineering Research Board (SERB), Department of Science and Technology, Government of India for the financial support through Project CRG/2020/002875.

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Correspondence to Árpád Baricz.

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Baricz, Á., Bisht, N., Singh, S. et al. Functional Inequalities and Bounds for the Generalized Marcum Function of the Second Kind. Results Math 76, 35 (2021). https://doi.org/10.1007/s00025-021-01343-3

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Keywords

  • Modified Bessel function of the second kind
  • bounds
  • log-convexity and geometrical convexity
  • survival function
  • complementary error function
  • generalized Marcum Q-function
  • symmetric difference

Mathematics Subject Classification

  • 33E20
  • 33C10
  • 94A13
  • 94A05
  • 26A48