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Modified Dini functions: monotonicity patterns and functional inequalities

Abstract

We deduce some new functional inequalities, like Turán type inequalities, Redheffer type inequalities, and a Mittag-Leffler expansion for a special combination of modified Bessel functions of the first kind, called modified Dini functions. Moreover, we show the complete monotonicity of a quotient of modified Dini functions by involving a new continuous infinitely divisible probability distribution. The key tool in our proofs is a recently developed infinite product representation for a special combination of Bessel functions of the first kind, which was very useful in determining the radius of convexity of some normalized Bessel functions of the first kind.

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Correspondence to Á. Baricz.

Additional information

The work of the first author was supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences.

The second author is on leave from the IIT Madras.

The research of the third author was supported by the fellowship of the University Grants Commission, India.

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Baricz, Á., Ponnusamy, S. & Singh, S. Modified Dini functions: monotonicity patterns and functional inequalities. Acta Math. Hungar. 149, 120–142 (2016). https://doi.org/10.1007/s10474-016-0599-9

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  • DOI: https://doi.org/10.1007/s10474-016-0599-9

Key words and phrases

  • functional inequality
  • Dini function
  • modified Bessel function
  • Turán type inequality
  • Redheffer type inequality
  • infinite product representation
  • completely monotonic
  • log-convex function

Mathematics Subject Classification

  • 39B62
  • 33C10
  • 42A05