Acta mathematica scientia, Series B >
THIRD-ORDER DIFFERENTIAL SUBORDINATION RESULTS FOR ANALYTIC FUNCTIONS INVOLVING THE GENERALIZED BESSEL FUNCTIONS
Received date: 2013-10-28
Revised date: 2014-02-14
Online published: 2014-11-20
Supported by
This work was partly supported by the Natural Science Foundation of China (11271045), the Higher School Doctoral Foundation of China (20100003110004), the Natural Science Foundation of Inner Mongolia of China (2010MS0117) and the Higher School Foundation of Inner Mongolia of China (NJZY13298) (to Tang); and by the Commission for the Scientific Research Projects of Kafkas Univertsity (2012-FEF-30) (to Deniz).
In the present paper, we derive some third-order differential subordination results for analytic functions in the open unit disk, using the operator Bcf by means of normalized form of the generalized Bessel functions of the first kind, which is defined as
z(BcK+1f(z))′ = KBcKf(z) − (K − 1)BcK+1f(z),
where b, c, p ∈ C and K = p + (b + 1)/2∈ C \ Z −0 (Z −0 = {0,−1,−2, · · · }). The results are obtained by considering suitable classes of admissible functions. Various known or new special cases of our main results are also pointed out.
TANG Huo , Erhan DENIZ . THIRD-ORDER DIFFERENTIAL SUBORDINATION RESULTS FOR ANALYTIC FUNCTIONS INVOLVING THE GENERALIZED BESSEL FUNCTIONS[J]. Acta mathematica scientia, Series B, 2014 , 34(6) : 1707 -1719 . DOI: 10.1016/S0252-9602(14)60116-8
[1] Ali R M, Ravichandran V, Seenivasagan N. Subordination and superordination on Schwarzian derivatives. J Ineq Appl, 2008, 2008: Article ID 12328
[2] Ali R M, Ravichandran V, Seenivasagan N. Subordination and superordination of the Liu-Srivastava linear operator on meromorphic functions. Bull Malays Math Sci Soc, 2008, 31(2): 193–207
[3] Ali R M, Ravichandran V, Seenivasagan N. Differential subordination and superordination of analytic functions defined by the multiplier transformation. Math Inequal Appl, 2009, 12(1): 123–139
[4] Ali R M, Ravichandran V, Seenivasagan N. Differential subordination and superordination of analytic functions defined by the Dziok-Srivastava operator. J Franklin Inst, 2010, 347(9): 1762–1781
[5] Ali R M, Ravichandran V, Seenivasagan N. On subordination and superordination of the multiplier trans-formation for meromorphic functions. Bull Malays Math Sci Soc, 2010, 33(2): 311–324
[6] Andras S, Baricz A. Monotonicity property of generalized and normalized Bessel functions of complex order. Complex Var Elliptic Equ, 2009, 54(7): 689–696
[7] Antonino J A, Miller S S. Third-order differential inequalities and subordinations in the complex plane. Complex Var Elliptic Equ, 2011, 56(5): 439–454
[8] Baricz A. Applications of the admissible functions method for some differential equations. Pure Appl Math, 2002, 13(4): 433–440
[9] Baricz A. Geometric properties of generalized Bessel functions. Publ Math Debrecen, 2008, 73: 155–178
[10] Baricz A. Generalized Bessel Functions of the First Kind. Lecture Notes in Mathematics, Vol 1994. Berlin: Springer-Verlag, 2010
[11] Baricz A, Ponnusamy S. Starlikeness and covexity of generalized Bessel function. Integral Transform Spec Funct, 2010, 21(9): 641–651
[12] Cho N E, Kwon O S, Owa S, Srivastava H M. A class of integral operators preserving subordination and superordination for meromorphic functions. Appl Math Comput, 2007, 193: 463–474
[13] Cho N E, Srivastava H M. A class of nonlinear integral operators preserving subordination and superordi-nation. Integral Transforms Spec Funct, 2007, 18: 95–107
[14] Deniz E. Convexity of integral operators involving generalized Bessel functions. Integral Transforms Spec Funct, 2013, 24(3): 201–216
[15] Deniz E, Orhan H, Srivastava H M. Some sufficient conditions for univalence of certain families of integral operators involving generalized Bessel functions. Taiwansese J Math, 2011, 15(2): 883–917
[16] Dziok J, Srivastava H M. Classes of analytic functions associated with the generalized hypergeometric function. Appl Math Comput, 1999, 103(1): 1–13
[17] Dziok J, Srivastava H M. Certain subclasses of analytic functions associated with the generalized hyperge-ometric function. Integral Transform Spec Funct, 2003, 14(1): 7–18
[18] Miller S S, Mocanu P T. Univalence of Gaussian and confluent hypergeometric functions. Proc Amer Math Soc, 1990, 110(2): 333–342
[19] Miller S S, Mocanu P T. Differential Subordinations. Monographs and Textbooks in Pure and Applied Mathematics, 225. New York: Marcel Dekker, 2000
[20] Owa S, Srivastava H M. Univalent and starlike generalized hypergeometric functions. Canad J Math, 1987, 39(5): 1057–1077
[21] Ponnusamy S, Juneja O P. Third-order differential inequalities in the complex plane. Current Topics in Analytic Function Theory. Singapore, London: World Scientific, 1992
[22] Ponnusamy S, Ronning F. Geometric properties for convolutions of hypergeometric functions and functions with the derivative in a halfplane. Integral Transform Spec Funct, 1999, 8: 121–138
[23] Ponnusamy S, Vuorinen M. Univalence and convexity properties for confluent hypergeometric functions. Complex Var Theory Appl, 1998, 36(1): 73–97
[24] Ponnusamy S, Vuorinen M. Univalence and convexity properties for Gaussian hypergeometric functions. Rocky Mountain J Math, 2001, 31(1): 327–353
[25] Prajapat J K. Certain geometric properties of normalized Bessel Functions. Appl Math Lett, 2011, 24: 2133–2139
[26] Selinger V. Geometric properties of normalized Bessel functions. Pure Math Appl, 1995, 6: 273–277
[27] Shanmugam T N, Sivasubramanian S, Srivastava H M. Differential sandwich theorems for certain subclasses of analytic functions involving multiplier transformations. Integral Transforms Spec Funct, 2006, 17: 889–899
[28] Srivastava H M, Yang D G, Xu N E. Subordinations for multivalent analytic functions associated with the Dziok-Srivastava operator. Integral Transforms Spec Funct, 2009, 20: 581–606
[29] Watson G N. A Treatise on the Theory of Bessel Functions. 2nd ed. Cambridge, London, New York: Cambridge University Press, 1944
/
| 〈 |
|
〉 |