Monotonic functions related to the q-gamma function

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• 作者：Ahmed Salem
• 关键词：Inequalities ; q ; Gamma function ; q ; Polygamma functions ; Monotonic function ; 33D05 ; 26D07 ; 26A48
• 刊名：Monatshefte f¨¹r Mathematik
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：179
• 期：2
• 页码：281-292
• 全文大小：399 KB
• 参考文献：1.Anderson, G.D., Qiu, S.-L.: A monotonicity property of the gamma function. Proc. Am. Math. Soc. 125, 3355–3362 (1997)MATH MathSciNet CrossRef
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  9.Salem, A.: A completely monotonic function involving \(q\) -gamma and \(q\) -digamma functions. J. Approx. Theory 164(7), 971–980 (2012)MATH MathSciNet CrossRef
  10.Abramowitz, M., Stegun, C.A.: Handbook of Mathematical Functions with Formulas, Graphs, Mathematical Tables 7th Printing, Applied Mathematics Series, vol. 55. Nathional Bureau of Standards, Washington, DC (1964)
  11.Krattenthaler, C., Srivastava, H.M.: Summations for basic hypergeometric series involving a \(q\) -analogue of the digamma function. Comput. Math. Appl. 32(2), 73–91 (1996)MATH MathSciNet CrossRef
  12.Salem, A.: Some Properties and expansions associated with the \(q\) -digamma function. Quaestiones Mathematicae 36(1), 67–77 (2013)MATH MathSciNet CrossRef
  13.Batir, N.: Monotonicity properties of \(q\) -digamma and \(q\) -trigamma functions. J. Approx. Theory 192(7), 336–346 (2015)MATH MathSciNet CrossRef
  14.Batir, N.: \(q\) -Extensions of some estimates associated with the digamma function. J. Approx. Theory 174, 54–64 (2013)MATH MathSciNet CrossRef
  15.Salem, A., Kamel, E.S.: Some completely monotonic functions associated with the \(q\) -gamma and the \(q\) -polygamma functions. Acta Math. Sci. 25(5), 1214–1224 (2015)MathSciNet CrossRef
  16.Salem, A.: Completely monotonic functions related to the \(q\) -gamma and the \(q\) -trigamma functions. Anal. Appl. 13(2), 125–134 (2015)MathSciNet CrossRef
  17.Salem, A.: On the \(q\) -beta function inequalities. Math. Inequ. Appl. 18(2), 639–648 (2015)MathSciNet
  18.Salem, A.: Two classes of bounds for the \(q\) -zeta functions. Banach J. Math. Anal. 8(1), 109–117 (2014)MATH MathSciNet CrossRef
  19.Salem, A.: Complete monotonicity properties of functions involving \(q\) -gamma and \(q\) -digamma functions. Math. Inequ. Appl. 17(3), 801–811 (2014)MATH MathSciNet
  20.Salem, A.: An infinite class of completely monotonic functions involving the \(q\) -gamma function. J. Math. Anal. Appl. 406(2), 392–399 (2013)MATH MathSciNet CrossRef
  21.Alzer, H., Grinshpan, A.Z.: Inequalities for the gamma and \(q\) -gamma functions. J. Approx. Theory 144, 67–83 (2007)MATH MathSciNet CrossRef
  
• 作者单位：Ahmed Salem (1)
  
  1. Department of Basic Science, Faculty of Information Systems and Computer Science, October 6 University, Sixth of October City, Egypt
  
• 刊物主题：Mathematics, general;
• 出版者：Springer Vienna
• ISSN：1436-5081

文摘

In this paper, the monotonicity property for two functions involving the logarithmic of the q-gamma function is proven for all \(q>0\). As a consequence, sharp inequalities for the q-gamma function are established. Our results are shown to be as a generalization of results which were obtained by Anderson and Qiu (Proc Am Math Soc 125:3355–3362, 1997). Keywords Inequalities q-Gamma function q-Polygamma functions Monotonic function