SUBORDINATION CONDITIONS FOR A CLASS OF NON-BAZILEVIČ TYPE DEFINED BY USING FRACTIONAL Q-CALCULUS OPERATORS

S. Abelman, K. A. Selvakumaran, M. M. Rashidi, S. D. Purohit

DOI Number
10.22190/FUMI1702255A
First page
255
Last page
267

Abstract


In this article, we introduce and investigate a new class of non-Bazilevič functions with respect to k-symmetric points defined by using fractional q-calculus operators and q-differentiation. Several interesting subordination results are derived for the functions belonging to this class in the open unit disc. Furthermore, we point out some new and known consequences of our main result.


Keywords

Fractional calculus; q-calculus; q-transform analysis.

Keywords


Univalent functions, starlike with respect to symmetric points, non-Bazilevivc functions, fractional q-calculus operators, subordination

Full Text:

PDF

References


bibitem{abu}M. H. Abu Risha, M. H. Annaby, M. E. H. Ismail and Z. S. Mansour, Linear $q$-difference equations, Z. Anal. Anwend.

{bf 26} (2007), no.~4, 481--494.

bibitem{alpu}D. Albayrak, S. D. Purohit and F. Uc{c}ar, On

$q$-analogues of Sumudu transforms, An. c{S}t. Univ. Ovidius

Constanc{t}a, {bf 21(1)}, (2013), 239--260.

bibitem{aldweby} H. Aldweby and M. Darus, A subclass of harmonic univalent functions associated with $q$-analogue of Dziok-Srivastava operator,

ISRN Mathematical Analysis, {bf 2013} (2013) Article ID 382312, 6 (pp).

bibitem{alo1}F. M. Al-Oboudi, On univalent functions defined by a generalized Su alu agean operator,

Int. J. Math. Math. Sci. {bf 2004}, no.~25-28, 1429--1436.

bibitem {BM}D. Baleanu and O. G. Mustafa, On the global existence of solutions

to a class of fractional differential equations, Comput. Math. Appl., 59(5) (2010), 1835--1841.

bibitem {BMA}D. Baleanu, O. G. Mustafa and R. P. Agarwal, On the solution

set for a class of sequential fractional differential equations, J. Phys. A Math. Theor., 43(38)(2010), Article ID 385209.

bibitem {BDST} D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, {it Fractional

Calculus Models and Numerical Methods}, Vol. 3 of Complexity, Nonlinearity and Chaos, World Scientific, 2012.

bibitem{banger}G. Bangerezako, Variational calculus on $q$-nonuniform lattices, J. Math. Anal. Appl. {bf 306} (2005), no.~1, 161--179.

bibitem{ernst}T. Ernst, {it A comprehensive treatment of $q$-calculus}, Birkh"auser/Springer Basel AG, Basel, 2012.

bibitem{gasper}G. Gasper and M. Rahman, {it Basic hypergeometric series}, Encyclopedia of Mathematics and its Applications, 35,

Cambridge Univ. Press, Cambridge, 1990.

bibitem{hall}D. J. Hallenbeck and S. Ruscheweyh, Subordination by convex functions,

Proc. Amer. Math. Soc. {bf 52} (1975), 191--195.

bibitem{jack}F. J. Jackson, On $q$-definite integrals, Q. J. Pure Appl. Math.

{bf 41} (1910), 193--203.

bibitem{kac}V. Kac and P. Cheung, {it Quantum calculus}, Universitext, Springer, New York, 2002.

bibitem{kumar} S. Kumar, D. Kumar, S. Abbasbandy and M. M. Rashidi, Analytical solution of fractional Navier-Stokes equation by using modified Laplace decomposition method, Ain Shams Engineering Journal, {bf 5(2)}, (2014), 569--574.

bibitem{man}Z. S. I. Mansour, Linear sequential $q$-difference equations of fractional order, Fract. Calc. Appl. Anal. {bf 12} (2009), no~2, 159--178.

bibitem{miller} S. S. Miller and P. T. Mocanu, Differential Subordinations: Theory and Applications,

Series in Pure and Applied Mathematics, No~225, Marcel Dekker, New York, (2000).

bibitem{murugu} G. Murugusundaramoorthy, C. Selvaraj and O. S. Babu, Subclasses of starlike functions associated with fractional

$q$-calculus operators, Journal of Complex Analysis, {bf 2013} (2013) Article ID 572718, 8 (pp).

bibitem{obra}M. Obradovi'c, {it A class of univalent functions}, Hokkaido Math. J., {bf 27(2)}, (1998), 329--335.

bibitem {PU} S. D. Purohit, Solutions of fractional partial differential equations

of quantum mechanics, Adva. Appl. Math. Mech., {bf 5(5)}, (2013), 639--651.

bibitem {PK}S. D. Purohit and S. L. Kalla, On fractional partial differential

equations related to quantum mechanics, J. Phys. A Math. Theor., {bf 44(4)}, (2011), Article ID 045202.

bibitem{purohit}S. D. Purohit and R. K. Raina, Certain subclasses of analytic functions associated with fractional $q$-calculus operators,

Math. Scand. {bf 109(1)} (2011), 55--70.

bibitem{purohit1} S. D. Purohit, A new class of multivalently analytic functions associated with fractional $q$-calculus operators,

Fractional Differ. Calc., {bf{2(2)}} (2012), 129--138.

bibitem{purohit3} S. D. Purohit and R. K. Raina, Fractional $q$-calculus and certain subclass of univalent analytic functions, Mathematica (Cluj), {bf 55(78)(1)}, (2013), 62--74.

bibitem{purohit4} S. D. Purohit and R. K. Raina, Some classes of analytic and multivalent functions associated with $q$-derivative operators,

Acta Universitatis Sapientiae, Mathematica, {bf 6(1)} (2014), 5--23.

bibitem{purohit5} S. D. Purohit and R. K. Raina, On a subclass of $p$-valent analytic functions involving fractional $q$-calculus operators,

Kuwait J. Sci., {bf 42(1)} (2015), 1--15.

bibitem{raj}P. M. Rajkovi'c, S. D. Marinkovi'c and M. S. Stankovi'c,

Fractional integrals and derivatives in $q$-calculus, Appl. Anal. Discrete Math. {bf 1} (2007), no.~1, 311--323.

bibitem{saka} K. Sakaguchi, On a certain univalent mapping, J. Math. Soc. Japan {bf 11} (1959), 72--75.

bibitem{sal}G. c S. Su alu agean, Subclasses of univalent functions, in {it Complex analysis---fifth

Romanian-Finnish seminar, Part 1 (Bucharest, 1981)}, 362--372,

Lecture Notes in Math., 1013, Springer, Berlin.

bibitem{selvakumaran1} K. A. Selvakumaran, S. D. Purohit, Aydin Secer and Mustafa Bayram, Convexity of certain $q$-integral operators of

$p$-valent functions, Abstract and Applied Analysis, {bf 2014}

(2014), Article ID 925902, 7 (pp).

bibitem{selvakumaran2} K. A. Selvakumaran, S. D. Purohit and Aydin

Secer, Majorization for a class of analytic functions defined by $q$

differentiation, Mathematical Problems in Engineering, {bf 2014}

(2014), Article ID 653917, 5 (pp).

bibitem{sel}C. Selvaraj and K. A. Selvakumaran, Subordination results for a class of non-Bazileviv c functions with

respect to symmetric points, Acta Univ. Apulensis Math. Inform. No.

(2011), 83--90.

bibitem{suff}T. J. Suffridge, Some remarks on convex maps of the unit disk, Duke Math. J. {bf 37} (1970), 775--777.

bibitem{tuneski}N. Tuneski and M. Darus, Fekete-SzegH o functional for non-Bazileviv c functions, Acta Math. Acad. Paedagog. Nyh'azi. (N.S.)

{bf 18(2)}, (2002), 63--65 (electronic).

bibitem{wang1}Z. Wang, C. Gao and M. Liao, On certain generalized

class of non-Bazileviv c functions, Acta Math. Acad. Paedagog.

Nyh'azi. (N.S.) {bf 21(2)}, (2005), 147--154 (electronic).

bibitem{yang} A. M. Yang, Y. Z. Zhang, C. Cattani, G. N. Xie, M. M. Rashidi, Y. J. Zhou and X. J. Yang, Application of local fractional series expansion method to solve Klein-Gordon equations on cantor sets, Abstract and Applied Analysis, {bf 2014}(2014) Article ID 372741, 6 (pp).




DOI: https://doi.org/10.22190/FUMI1702255A

Refbacks

  • There are currently no refbacks.




© University of Niš | Created on November, 2013
ISSN 0352-9665 (Print)
ISSN 2406-047X (Online)