Document Type : Research Paper
Authors
Department of Mathematics, Payame Noor University (PNU), P.O.Box: 19395-3697, Tehran, Iran.
Abstract
In this paper, by making use of $(p , q) $-derivative operator we introduce a new subclass of meromorphically univalent functions. Precisely, we give a necessary and sufficient coefficient condition for functions in this class. Coefficient estimates, extreme points, convex linear combination, Radii of starlikeness and convexity and finally partial sum property are investigated.
Keywords
[1] M.H. Abu-Risha, M.H. Annaby, M.E.H. Ismail and Z.S. Mansour, Linear $q$-difference equations, Z. Anal. Anwend., 26 (4) (2007), pp. 481-494.
[2] D. Albayrak, S.D. Purohit and F. Uçar, On $q$-analogues of sumudu transforms, An. Ştiinţ. Univ. Ovidius Constanţa Ser. Mat., 21 (1) (2013), pp. 239-260.
[3] R. Chakrabarti, R. Jagannathan, A $(p,q) $-oscillator realization of two-parameter quantum algebras, J. Phys. A., 24 (1991), pp. L711-L718.
[4] G. Gasper and M. Rahman, Basic Hypergeometric series, Cambridge University Press, Cambridge, 1990.
[5] Z.-G. Wang and M.-L. Li, Some properties of Certain family of Multiplier transforms, Filomat., 31 (1) (2017), pp. 159-173.
[6] F.H. Jackson, On $q$-functions and a certain difference operator, Trans. Roy. Soc. Edinb., 46 (1908), pp. 64-72.
[7] F.H. Jackson, $q$-definite integrals, Q. J. pure Appl. Math., 41 (1910), pp. 193-203.
[8] F.H. Jackson, $q$-Difference equations, Am. J. Math., 32 (1910), pp. 305-314.
[9] Z.S.I. Mansour, Linear sequential $q$-difference equations of fractional order, Fract. Calc. Appl. Anal., 12 (2) (2009), pp. 159-178.
[10] A.O. Mostafa, M.K. Aouf, H.M. Zayed and T. Bulboaca, Convolution conditions for subelasse of mermorphic functions of complex order associated with basic Bessel functions, J. Egyptian Math. Soc., 25 (2017), pp. 286-290.
[11] H.E. Ozkan Ucar, Coefficient inequalties for $q$-starlike functions, Appl. Math. Comput., 276 (2016), pp. 122-126.
[12] Y. Polatoglu, Growth and distortion theorems for generalized $q$-starlike functions, Adv. Math. Sci. J., 5 (2016), pp. 7-12.
[13] S.D. Purohit and R.K. Raina, Certain subclass of analytic functions associated with fractional $q$-calculus operators, Math. Scand., 109 (2011), pp. 55-70.
[14] P.M. Rajkovic, S.D. Marinkovic and M.S. Stankovic, Fractional integrals and derivatives in $q$-calculus, Appl. Anal. Discrete Math., 1 (2007), pp. 311-323.
[15] R. Srivastava and H.M. Zayed, Subclasses of analytic functions of complex order definde by $q$-derivative operator, Stud. Univ. Babes-Bolyai Math., 64 (2019), pp. 71-80.
[16] M. Tahir, N. Khan, Q.Z. Ahmad, B. Khan and G.M. Khan, Coefficient Estimates for Some Subclasses of Analytic and Bi-Univalent Functions Associated with Conic Domain, Sahand Commun. Math. Anal., 16 (2019), pp. 69-81.
[17] A. Zireh and M.M. Shabani, On the Linear Combinations of Slanted Half-Plane Harmonic Mappings, Sahand Commun. Math. Anal., 14 (2019), pp. 89-96.
[2] D. Albayrak, S.D. Purohit and F. Uçar, On $q$-analogues of sumudu transforms, An. Ştiinţ. Univ. Ovidius Constanţa Ser. Mat., 21 (1) (2013), pp. 239-260.
[3] R. Chakrabarti, R. Jagannathan, A $(p,q) $-oscillator realization of two-parameter quantum algebras, J. Phys. A., 24 (1991), pp. L711-L718.
[4] G. Gasper and M. Rahman, Basic Hypergeometric series, Cambridge University Press, Cambridge, 1990.
[5] Z.-G. Wang and M.-L. Li, Some properties of Certain family of Multiplier transforms, Filomat., 31 (1) (2017), pp. 159-173.
[6] F.H. Jackson, On $q$-functions and a certain difference operator, Trans. Roy. Soc. Edinb., 46 (1908), pp. 64-72.
[7] F.H. Jackson, $q$-definite integrals, Q. J. pure Appl. Math., 41 (1910), pp. 193-203.
[8] F.H. Jackson, $q$-Difference equations, Am. J. Math., 32 (1910), pp. 305-314.
[9] Z.S.I. Mansour, Linear sequential $q$-difference equations of fractional order, Fract. Calc. Appl. Anal., 12 (2) (2009), pp. 159-178.
[10] A.O. Mostafa, M.K. Aouf, H.M. Zayed and T. Bulboaca, Convolution conditions for subelasse of mermorphic functions of complex order associated with basic Bessel functions, J. Egyptian Math. Soc., 25 (2017), pp. 286-290.
[11] H.E. Ozkan Ucar, Coefficient inequalties for $q$-starlike functions, Appl. Math. Comput., 276 (2016), pp. 122-126.
[12] Y. Polatoglu, Growth and distortion theorems for generalized $q$-starlike functions, Adv. Math. Sci. J., 5 (2016), pp. 7-12.
[13] S.D. Purohit and R.K. Raina, Certain subclass of analytic functions associated with fractional $q$-calculus operators, Math. Scand., 109 (2011), pp. 55-70.
[14] P.M. Rajkovic, S.D. Marinkovic and M.S. Stankovic, Fractional integrals and derivatives in $q$-calculus, Appl. Anal. Discrete Math., 1 (2007), pp. 311-323.
[15] R. Srivastava and H.M. Zayed, Subclasses of analytic functions of complex order definde by $q$-derivative operator, Stud. Univ. Babes-Bolyai Math., 64 (2019), pp. 71-80.
[16] M. Tahir, N. Khan, Q.Z. Ahmad, B. Khan and G.M. Khan, Coefficient Estimates for Some Subclasses of Analytic and Bi-Univalent Functions Associated with Conic Domain, Sahand Commun. Math. Anal., 16 (2019), pp. 69-81.
[17] A. Zireh and M.M. Shabani, On the Linear Combinations of Slanted Half-Plane Harmonic Mappings, Sahand Commun. Math. Anal., 14 (2019), pp. 89-96.
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