Title: Inequalities for the Modified k-Bessel Function
Author(s): Saiful Rahman Mondal, Kottakkaran Sooppy Nisar
Pages: 203-208
Cite as:
Saiful Rahman Mondal, Kottakkaran Sooppy Nisar, Inequalities for the Modified k-Bessel Function, Int. J. Anal. Appl., 14 (2) (2017), 203-208.

Abstract


The article considers the generalized k-Bessel functions and represents it as Wright functions. Then we study the monotonicity properties of the ratio of two different orders k- Bessel functions, and the ratio of the k-Bessel and the k-Bessel functions. The log-convexity with respect to the order of the k-Bessel also given. An investigation regarding the monotonicity of the ratio of the k-Bessel and k-confluent hypergeometric functions are discussed.

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References


  1. L.G. Romero, G.A.Dorrego and R.A. Cerutti, The k-Bessel function of first kind, Int. Math. Forum, 38(7)(2012), 1859–1854. Google Scholar

  2. GN. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Mathematical Library Edition, Cambridge University Press, Cambridge (1995). Reprinted (1996) Google Scholar

  3. A. Erd´ elyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Higher transcendental functions, I, II, McGraw-Hill Book Company, Inc., New York, 1953. New York, Toronto, London, 1953. Google Scholar

  4. R. Diaz and E. Pariguan, On hypergeometric functions and k-Pochhammer symbol, Divulg. Mat. 15(2) (2007), 179–192. Google Scholar

  5. K. Nantomah, E. Prempeh, Some Inequalities for the k-Digamma Function, Math. Aeterna, 4(5) (2014), 521–525. Google Scholar

  6. S. Mubeen, M. Naz and G. Rahman, A note on k-hypergemetric differential equations, J. Inequal. Spec. Funct. 4(3) (2013), 8–43. Google Scholar

  7. M. Biernacki and J. Krzy˙ z, On the monotonicity of certain functionals in the theory of analytic functions, Ann. Univ. Mariae Curie-Sk lodowska. Sect. A. 9 (1957), 135–147. Google Scholar

  8. C. G. Kokologiannaki, Properties and inequalities of generalized k-gamma, beta and zeta functions, Int. J. Contemp. Math. Sci. 5(13-16) (2010), 653–660. Google Scholar

  9. C. G. Kokologiannaki and V. Krasniqi, Some properties of the k-gamma function, Matematiche (Catania), 68(1) (2013), 13–22. Google Scholar

  10. V. Krasniqi, A limit for the k-gamma and k-beta function, Int. Math. Forum, 5(33-36) (2010), 1613–1617. Google Scholar

  11. M. Mansour, Determining the k-generalized gamma function Γ k (x) by functional equations, Int. J. Contemp. Math. Sci., 4(21-24) (2009), 1037–1042. Google Scholar

  12. G. E. Andrews, R. Askey and R. Roy, Special functions, Cambridge Univ. Press, Cambridge, 1999. Google Scholar

  13. C. Fox, The asymptotic expansion of generalized hypergeometric functions, Proc. London Math. Soc., 27(4) (1928), 389–400. Google Scholar

  14. A. A. Kilbas, M. Saigo and J. J. Trujillo, On the generalized Wright function, Fract. Calc. Appl. Anal. 5(4) (2002), 437–460. Google Scholar

  15. A. A. Kilbas and N. Sebastian, Generalized fractional integration of Bessel function of the first kind, Integral Transforms Spec. Funct. 19 (11-12) (2008), 869–883. Google Scholar

  16. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies 204, Elsevier, Amsterdam, 2006. Google Scholar

  17. E. D. Rainville, Special functions, Macmillan, New York, 1960. Google Scholar

  18. E. M. Wright, The asymptotic expansion of integral functions defined by Taylor series, Philos. Trans. Roy. Soc. London, Ser. A. 238 (1940), 423–451. Google Scholar

  19. E. M. Wright, The asymptotic expansion of the generalized hypergeometric function, Proc. London Math. Soc. (2) 46(1940), 389–408. Google Scholar