• Title, Summary, Keyword: Kershaw's inequality

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A CLASS OF COMPLETELY MONOTONIC FUNCTIONS INVOLVING DIVIDED DIFFERENCES OF THE PSI AND TRI-GAMMA FUNCTIONS AND SOME APPLICATIONS

  • Guo, Bai-Ni;Qi, Feng
    • Journal of the Korean Mathematical Society
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    • v.48 no.3
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    • pp.655-667
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    • 2011
  • A class of functions involving divided differences of the psi and tri-gamma functions and originating from Kershaw's double inequality are proved to be completely monotonic. As applications of these results, the monotonicity and convexity of a function involving the ratio of two gamma functions and originating from the establishment of the best upper and lower bounds in Kershaw's double inequality are derived, two sharp double inequalities involving ratios of double factorials are recovered, the probability integral or error function is estimated, a double inequality for ratio of the volumes of the unit balls in $\mathbb{R}^{n-1}$ and $\mathbb{R}^n$ respectively is deduced, and a symmetrical upper and lower bounds for the gamma function in terms of the psi function is generalized.

FOUR LOGARITHMICALLY COMPLETELY MONOTONIC FUNCTIONS INVOLVING GAMMA FUNCTION

  • Qi, Feng;Niu, Da-Wei;Cao, Jian;Chen, Shou-Xin
    • Journal of the Korean Mathematical Society
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    • v.45 no.2
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    • pp.559-573
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    • 2008
  • In this paper, two classes of functions, involving a parameter and the classical Euler gamma function, and two functions, involving the classical Euler gamma function, are verified to be logarithmically completely monotonic in $(-\frac{1}{2},\infty)$ or $(0,\infty)$; some inequalities involving the classical Euler gamma function are deduced and compared with those originating from certain problems of traffic flow, due to J. Wendel and A. Laforgia, and relating to the well known Stirling's formula.