Inequalities for the Modified k-Bessel Function

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Saiful Rahman Mondal, Kottakkaran Sooppy Nisar


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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  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.
  2. GN. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Mathematical Library Edition, Cambridge University Press, Cambridge (1995). Reprinted (1996)
  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.
  4. R. Diaz and E. Pariguan, On hypergeometric functions and k-Pochhammer symbol, Divulg. Mat. 15(2) (2007), 179-192.
  5. K. Nantomah, E. Prempeh, Some Inequalities for the k-Digamma Function, Math. Aeterna, 4(5) (2014), 521-525.
  6. S. Mubeen, M. Naz and G. Rahman, A note on k-hypergemetric differential equations, J. Inequal. Spec. Funct. 4(3) (2013), 8-43.
  7. M. Biernacki and J. KrzyË™ 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.
  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.
  9. C. G. Kokologiannaki and V. Krasniqi, Some properties of the k-gamma function, Matematiche (Catania), 68(1) (2013), 13-22.
  10. V. Krasniqi, A limit for the k-gamma and k-beta function, Int. Math. Forum, 5(33-36) (2010), 1613-1617.
  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.
  12. G. E. Andrews, R. Askey and R. Roy, Special functions, Cambridge Univ. Press, Cambridge, 1999.
  13. C. Fox, The asymptotic expansion of generalized hypergeometric functions, Proc. London Math. Soc., 27(4) (1928), 389-400.
  14. A. A. Kilbas, M. Saigo and J. J. Trujillo, On the generalized Wright function, Fract. Calc. Appl. Anal. 5(4) (2002), 437-460.
  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.
  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.
  17. E. D. Rainville, Special functions, Macmillan, New York, 1960.
  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.
  19. E. M. Wright, The asymptotic expansion of the generalized hypergeometric function, Proc. London Math. Soc. (2) 46(1940), 389-408.