Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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MONOTONICITY PROPERTIES OF THE GENERALIZED STRUVE FUNCTIONS

  • Ali, Rosihan M. (School of Mathematical Sciences Universiti Sains Malaysia) ;
  • Mondal, Saiful R. (Department of Mathematics and Statistics Collage of Science King Faisal University) ;
  • Nisar, Kottakkaran S. (Department of Mathematics College of Arts and Science Prince Sattam bin Abdulaziz University)
  • Received : 2016.03.01
  • Published : 2017.03.01
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Abstract

This paper introduces and studies a generalization of the classical Struve function of order p given by $$_aS_{p,c}(x):=\sum\limits_{k=0}^{\infty}\frac{(-c)^k}{{\Gamma}(ak+p+\frac{3}{2}){\Gamma}(k+\frac{3}{2})}(\frac{x}{2})^{2k+p+1}$$. Representation formulae are derived for $_aS_{p,c}$. Further the function $_aS_{p,c}$ is shown to be a solution of an (a + 1)-order differential equation. Monotonicity and log-convexity properties for the generalized Struve function $_aS_{p,c}$ are investigated, particulary for the case c = -1. As a consequence, $Tur{\acute{a}}n$-type inequalities are established. For a = 2 and c = -1, dominant and subordinant functions are obtained for the Struve function $_2S_{p,-1}$.

Keywords

Acknowledgement

Supported by : FRGS

References

  1. M. Abramowitz and I. A. Stegun, A Handbook of Mathematical Functions, New York, 1965.
  2. A. R. Ahmadi and S. E. Widnall, Unsteady lifting-line theory as a singular-perturbation problem, J. Fluid Mech. 153 (1985), 59-81. https://doi.org/10.1017/S0022112085001148
  3. G. E. Andrews, R. Askey, and R. Roy, Special functions, Encyclopedia of Mathematics and its Applications, 71, Cambridge Univ. Press, Cambridge, 1999.
  4. A. Baricz, Functional inequalities for Galue's generalized modified Bessel functions, J. Math. Inequal. 1 (2007), no. 2, 183-193.
  5. A. Baricz and T. K. Pogany, Integral representations and summations of the modified Struve function, Acta Math. Hungar. 141 (2013), no. 3, 254-281. https://doi.org/10.1007/s10474-013-0308-x
  6. A. Baricz and T. K. Pogany, Functional inequalities for modified Struve functions, Proc. Roy. Soc. Edinburgh Sect. A 144 (2014), no. 5, 891-904. https://doi.org/10.1017/S0308210512001370
  7. M. Biernacki and J. Krzyz, On the monotonity of certain functionals in the theory of analytic functions, Ann. Univ. Mariae Curie-Sk lodowska. Sect. A. 9 (1955), 135-147.
  8. K. N. Bhowmick, Some relations between a generalized Struve's function and hyperge-ometric functions, Vijnana Parishad Anusandhan Patrika 5 (1962), 93-99.
  9. K. N. Bhowmick, A generalized Struve's function and its recurrence formula, Vijnana Parishad Anusandhan Patrika 6 (1963), 1-11.
  10. K. N. Bhowmick, Some integrals involving a generalized Struve's function, J. Sci. Res. Banaras Hindu Univ. 13 (1962/1963), 244-251.
  11. S. B. Choudhary and G. S. Prasad, Some pure recurrence relations for generalized Struve function $H^{\lambda}_{{\nu},{\alpha},{\beta}}$(z), Ranchi Univ. Math. J. 16 (1985), 94-99.
  12. L. Galue, A generalized Bessel function, Integral Transforms Spec. Funct. 14 (2003), no. 5, 395-401. https://doi.org/10.1080/1065246031000074362
  13. M. H. Hirata, Flow near the bow of a steadily turning ship, J. Fluid Mech. 71 (1975), no. 2, 283-291. https://doi.org/10.1017/S0022112075002571
  14. C. M. Joshi, Some integrals involving a generalized Struve function, Vijnana Parishad Anusandhan Patrika 10 (1967), 137-145.
  15. C. M. Joshi and S. Nalwaya, Inequalities for modified Struve functions, J. Indian Math. Soc. (N.S.) 65 (1998), no. 1-4, 49-57.
  16. B. N. Kanth, Integrals involving generalised Struve's function, Nepali Math. Sci. Rep. 6 (1981), no. 1-2, 61-64.
  17. A. Laforgia and P. Natalini, Turan-type inequalities for some special functions, J. Inequal. Pure Appl. Math. 7 (2006), no. 1, Article 22, 3 pp.
  18. D. S. Mitrinovic, Analytic inequalities, Springer, New York, 1970.
  19. F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST handbook of mathematical functions, U.S. Dept. Commerce, Washington, DC, 2010.
  20. H. Orhan and N. Yagmur, Geometric Properties of Generalized Struve Functions, An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.), DOI: 10.2478/aicu-2014-0007, May 2014.
  21. T. G. Pedersen, Variational approach to excitons in carbon nanotubes, Phys. Rev. B 67 (2003), no. 7, 073401. https://doi.org/10.1103/PhysRevB.67.073401
  22. J. Shao and P. Hanggi, Decoherent dynamics of a two-level system coupled to a sea of spins, Phys. Rev. Lett. 81 (1998), no. 26, 5710-5713. https://doi.org/10.1103/PhysRevLett.81.5710
  23. D. C. Shaw, Perturbational results for diffraction of water-waves by nearly-vertical barriers, IMA, J. Appl. Math. 34 (1985), no. 1, 99-117. https://doi.org/10.1093/imamat/34.1.99
  24. R. P. Singh, Generalized Struve's function and its recurrence relations, Ranchi Univ. Math. J. 5 (1974), 67-75.
  25. R. P. Singh, Generalized Struve's function and its recurrence equation, Vijnana Parishad Anusandhan Patrika 28 (1985), no. 3, 287-292.
  26. R. P. Singh, Some integral representation of generalized Struve's function, Math. Ed. (Siwan) 22 (1988), no. 3, 91-94.
  27. R. P. Singh, On definite integrals involving generalized Struve's function, Math. Ed. (Siwan) 22 (1988), no. 2, 62-66.
  28. R. P. Singh, Infinite integrals involving generalized Struve function, Math. Ed. (Siwan) 23 (1989), no. 1, 30-36.
  29. N. Yagmur and H. Orhan, Starlikeness and convexity of generalized Struve functions, Abstr. Appl. Anal. 2013, Art. ID 954513, 6 pp.
  30. N. Yagmur and H. Orhan, Hardy space of generalized Struve functions, Complex Var. Elliptic Equ. 59 (2014), no. 7, 929-936. https://doi.org/10.1080/17476933.2013.799148

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