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A Survey on Cauchy–Bunyakovsky–Schwarz Inequality for Power Series

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Abstract

In this paper, we present a survey of some recent results for the celebrated Cauchy–Bunyakovsky–Schwarz inequality for functions defined by power series with nonnegative coefficients. Particular examples for fundamental functions of interest are presented. Applications for some special functions are given as well.

Keywords

  • Power Series
  • Hypergeometric Function
  • Product Space
  • Schwarz Inequality
  • Modify Bessel Function

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References

  1. Abramowitz, M., Stegun, I. (eds.): Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Dover, New York (1972)

    MATH  Google Scholar 

  2. Alzer, H.: A refinement of the Cauchy-Schwarz inequality. J. Math. Anal. Appl. 168, 596–604 (1992)

    CrossRef  MATH  MathSciNet  Google Scholar 

  3. Buzano, M.L.: Generalizzazione della diseguaglianza di Cauchy-Schwarz (Italian). Rend. Sem. Mat. Univ. e Politech. Torino. 31(1971/73), 405–409 (1974)

    MathSciNet  Google Scholar 

  4. Cerone, P., Dragomir, S.S.: Some applications of de Bruijn’s inequality for power series. Integral Transform. Spec. Funct. 18(6), 387–396 (2007)

    CrossRef  MATH  MathSciNet  Google Scholar 

  5. de Bruijn, N.G.: Problem 12. Wisk. Opgaven 21, 12–14 (1960)

    Google Scholar 

  6. Dragomir, S.S.: Some refinements of Schwarz inequality. In: Simpozionul de Mathematişi Aplicaţii, Timişoara, pp. 13–16, Romania, 1–2 Noiembrie 1985

    Google Scholar 

  7. Dragomir, S.S.: Inequalities of Cauchy–Bunyakovsky–Schwarz’s type for positive linear functionals (Romanian). Gaz. Mat. Metod. (Bucharest) 9, 162–164 (1988)

    Google Scholar 

  8. Dragomir, S.S.: Some refinements of Cauchy-Bunyakovsky-Schwarz inequality for sequences. In: Proceedings of the Third Symposium of Mathematics and Its Applications, pp. 78–82 (1989)

    Google Scholar 

  9. Dragomir, S.S.: A survey on Cauchy–Bunyakovsky–Schwarz Type discrete inequalities. J. Inequal. Pure Appl. Math. 4(3, 63), 1–140 (2003)

    Google Scholar 

  10. Dragomir, S.S.: Discrete Inequalities of the Cauchy Bunyakovsky Schwarz Type. Nova Science Publishers Inc., New York (2004)

    MATH  Google Scholar 

  11. Dragomir, S.S.: Advances in Inequality of the Schwarz, Grüss and Bessel Type in Inner Product Spaces. Nova Science Publishers Inc., New York (2005)

    Google Scholar 

  12. Dragomir, S.S.: Advances in Inequality of the Schwarz, Triangle and Heisenberg Type in Inner Product Space. Nova Science Publishers Inc., New York (2007)

    Google Scholar 

  13. Ibrahim, A., Dragomir, S.S.: Power series inequalities via Buzano’s result and applications. Integral Transform. Spec. Funct. 22(12), 867–878 (2011)

    CrossRef  MATH  MathSciNet  Google Scholar 

  14. Ibrahim, A., Dragomir, S.S.: Power series inequalities via a refinement of Schwarz inequality. Integral Transform. Spec. Funct. 23(10), 769–781 (2012)

    CrossRef  MATH  MathSciNet  Google Scholar 

  15. Ibrahim, A., Dragomir, S.S., Darus, M.: Some inequalities for power series with applications. Integral Transform. Spec. Funct. 24(5), 364–376 (2013)

    CrossRef  MATH  MathSciNet  Google Scholar 

  16. Mitrinović, D.S., Pečarić, J.E., Fink, A.M.: Classical and New Inequalities in Analysis. Kluwer Academic Publishers, Dordrecht (1993)

    CrossRef  MATH  Google Scholar 

  17. Zheng, L.: Remark on a refinement of the Cauchy–Schwarz inequality. J. Math. Anal. Appl. 218, 13–21 (1998)

    CrossRef  MATH  MathSciNet  Google Scholar 

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Correspondence to Alawiah Ibrahim .

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Dedicated to Professor Hari M. Srivastava

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Ibrahim, A., Dragomir, S.S. (2014). A Survey on Cauchy–Bunyakovsky–Schwarz Inequality for Power Series. In: Milovanović, G., Rassias, M. (eds) Analytic Number Theory, Approximation Theory, and Special Functions. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0258-3_10

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