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$q$-Analogue of a New Subclass of Harmonic Univalent Functions Defined by Fractional Calculus Operator

Year 2021, Volume: 9 Issue: 2, 42 - 52, 01.06.2021

Abstract

The purpose of the present paper is to introduce a new subclass of harmonic univalent functions by using fractional calculus operator associated with $q$-calculus. Coefficient condition, extreme points, distortion bounds, convolution and convex combination are obtained for this class. Finally, we discuss a class preserving integral operator for this class.

References

  • [1] Arif, M., Barkub, O., Srivastava, H.M., Abdullah, S., Khan, S.A.: Some Janowski type harmonic q-starlike functions associated with symmetrical points. Mathematics. 8, Art. 629, 1-18 (2020).
  • [2] Ahuja, O.P., Centinkaya, A. and Ravichandran, V.: Harmonic univalent functions defined by post quantum calculus operators. Acta Univ. Sapientiae, Mathematica. 11(1), 5–17 (2019).
  • [3] Clunie, J., Sheil-Small, T.: Harmonic univalent functions. Ann. Acad. Sci. Fen. Series AI Math. 9 (3), 3-25 (1984).
  • [4] Dixit, K.K., Porwal, S.: A new subclass of harmonic univalent functions defined by fractional calculus operator. General Math. 19 (2), 81-89 (2011).
  • [5] Duren, P.: Harmonic Mappings in the Plane, Cambridge Tracts in Mathematics, Vol.156, Cambridge University Press, Cambridge,(2004).
  • [6] Jackson, F.H.: On q-definite integrals., Quart. J. Pure Appl. Math. 41, 193–203 (1910).
  • [7] Jackson, F.H.:, q-difference equations., Am. J. Math. 32, 305–314 (1910).
  • [8] Jahangiri, J.M.: Harmonic univalent functions defined by q-calculus operators. Int. J. Math. Anal. Appl. 5 (2), 39–43 (2018).
  • [9] Jahangiri, J.M., Kim, Y.C., Srivastava, H.M.: Construction of a certain class of harmonic close-to-convex functions associated with the Alexander integral transform. Integral Trans. Special Funct. 14 (3), 237–242 (2003).
  • [10] Najafzadeh, Sh., Makinde, D.O.: Certain subfamily of harmonic functions related to Salagean q-differential operator. Int. J. Anal. Appl. 18(2), 254-261 (2020).
  • [11] Owa, S.: On the distortion theorem I. Kyungpook Math. J. 18, 53-59 (1978).
  • [12] Porwal, S., Gupta, A.: An application of q- calculus to harmonic univalent functions. J. Qual. Measurement Anal. 14(1) , 81-90 (2018).
  • [13] Porwal, S., Aouf, M.K.: On a new subclass of harmonic univalent functions defined by fractional calculus operator. J. Frac. Calc. Appl. 4(10), 1-12 (2013).
  • [14] Ravindar, B., Sharma, R. B., Magesh, N.: On a subclass of harmonic univalent functions defined by Ruscheweyh q- differential operator. AIP Conference Proceedings. 2112(1) , 1-12 (2019).
  • [15] Ravindar, B., Sharma, R. B., Magesh, N.: On a certain subclass of harmonic univalent functions defined q-differential operator. J. Mech. Cont. Math. Sci. 14(6), 45-53 (2019).
  • [16] Salagean, G.S.: Subclasses of univalent functions, Complex Analysis-Fifth Romanian Finish Seminar, Bucharest. 1 , 362-372 (1983).
  • [17] Srivastava, H.M.: Operators of basic (or q􀀀) calculus and fractional q-calculus and their applications in geometric function theory of complex analysis. Iran. J. Sci. Technol. Trans. A: Sci. 44 , 327-344 (2020).
  • [18] Srivastava, H.M., Owa, S.: An application of the fractional derivative. Math. Japon. 29 , 383-389 (1984).
  • [19] Uralegaddi, B.A., Ganigi, M.D., Sarangi, S.M.: Close-to-Convex functions with positive coefficients. Studia Univ. Babes-Balyai Math. XL 4, 25-31 (1995).
Year 2021, Volume: 9 Issue: 2, 42 - 52, 01.06.2021

Abstract

References

  • [1] Arif, M., Barkub, O., Srivastava, H.M., Abdullah, S., Khan, S.A.: Some Janowski type harmonic q-starlike functions associated with symmetrical points. Mathematics. 8, Art. 629, 1-18 (2020).
  • [2] Ahuja, O.P., Centinkaya, A. and Ravichandran, V.: Harmonic univalent functions defined by post quantum calculus operators. Acta Univ. Sapientiae, Mathematica. 11(1), 5–17 (2019).
  • [3] Clunie, J., Sheil-Small, T.: Harmonic univalent functions. Ann. Acad. Sci. Fen. Series AI Math. 9 (3), 3-25 (1984).
  • [4] Dixit, K.K., Porwal, S.: A new subclass of harmonic univalent functions defined by fractional calculus operator. General Math. 19 (2), 81-89 (2011).
  • [5] Duren, P.: Harmonic Mappings in the Plane, Cambridge Tracts in Mathematics, Vol.156, Cambridge University Press, Cambridge,(2004).
  • [6] Jackson, F.H.: On q-definite integrals., Quart. J. Pure Appl. Math. 41, 193–203 (1910).
  • [7] Jackson, F.H.:, q-difference equations., Am. J. Math. 32, 305–314 (1910).
  • [8] Jahangiri, J.M.: Harmonic univalent functions defined by q-calculus operators. Int. J. Math. Anal. Appl. 5 (2), 39–43 (2018).
  • [9] Jahangiri, J.M., Kim, Y.C., Srivastava, H.M.: Construction of a certain class of harmonic close-to-convex functions associated with the Alexander integral transform. Integral Trans. Special Funct. 14 (3), 237–242 (2003).
  • [10] Najafzadeh, Sh., Makinde, D.O.: Certain subfamily of harmonic functions related to Salagean q-differential operator. Int. J. Anal. Appl. 18(2), 254-261 (2020).
  • [11] Owa, S.: On the distortion theorem I. Kyungpook Math. J. 18, 53-59 (1978).
  • [12] Porwal, S., Gupta, A.: An application of q- calculus to harmonic univalent functions. J. Qual. Measurement Anal. 14(1) , 81-90 (2018).
  • [13] Porwal, S., Aouf, M.K.: On a new subclass of harmonic univalent functions defined by fractional calculus operator. J. Frac. Calc. Appl. 4(10), 1-12 (2013).
  • [14] Ravindar, B., Sharma, R. B., Magesh, N.: On a subclass of harmonic univalent functions defined by Ruscheweyh q- differential operator. AIP Conference Proceedings. 2112(1) , 1-12 (2019).
  • [15] Ravindar, B., Sharma, R. B., Magesh, N.: On a certain subclass of harmonic univalent functions defined q-differential operator. J. Mech. Cont. Math. Sci. 14(6), 45-53 (2019).
  • [16] Salagean, G.S.: Subclasses of univalent functions, Complex Analysis-Fifth Romanian Finish Seminar, Bucharest. 1 , 362-372 (1983).
  • [17] Srivastava, H.M.: Operators of basic (or q􀀀) calculus and fractional q-calculus and their applications in geometric function theory of complex analysis. Iran. J. Sci. Technol. Trans. A: Sci. 44 , 327-344 (2020).
  • [18] Srivastava, H.M., Owa, S.: An application of the fractional derivative. Math. Japon. 29 , 383-389 (1984).
  • [19] Uralegaddi, B.A., Ganigi, M.D., Sarangi, S.M.: Close-to-Convex functions with positive coefficients. Studia Univ. Babes-Balyai Math. XL 4, 25-31 (1995).
There are 19 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Saurabh Porwal 0000-0003-0847-3550

Publication Date June 1, 2021
Submission Date April 11, 2020
Acceptance Date February 15, 2021
Published in Issue Year 2021 Volume: 9 Issue: 2

Cite

APA Porwal, S. (2021). $q$-Analogue of a New Subclass of Harmonic Univalent Functions Defined by Fractional Calculus Operator. Mathematical Sciences and Applications E-Notes, 9(2), 42-52.
AMA Porwal S. $q$-Analogue of a New Subclass of Harmonic Univalent Functions Defined by Fractional Calculus Operator. Math. Sci. Appl. E-Notes. June 2021;9(2):42-52.
Chicago Porwal, Saurabh. “$q$-Analogue of a New Subclass of Harmonic Univalent Functions Defined by Fractional Calculus Operator”. Mathematical Sciences and Applications E-Notes 9, no. 2 (June 2021): 42-52.
EndNote Porwal S (June 1, 2021) $q$-Analogue of a New Subclass of Harmonic Univalent Functions Defined by Fractional Calculus Operator. Mathematical Sciences and Applications E-Notes 9 2 42–52.
IEEE S. Porwal, “$q$-Analogue of a New Subclass of Harmonic Univalent Functions Defined by Fractional Calculus Operator”, Math. Sci. Appl. E-Notes, vol. 9, no. 2, pp. 42–52, 2021.
ISNAD Porwal, Saurabh. “$q$-Analogue of a New Subclass of Harmonic Univalent Functions Defined by Fractional Calculus Operator”. Mathematical Sciences and Applications E-Notes 9/2 (June 2021), 42-52.
JAMA Porwal S. $q$-Analogue of a New Subclass of Harmonic Univalent Functions Defined by Fractional Calculus Operator. Math. Sci. Appl. E-Notes. 2021;9:42–52.
MLA Porwal, Saurabh. “$q$-Analogue of a New Subclass of Harmonic Univalent Functions Defined by Fractional Calculus Operator”. Mathematical Sciences and Applications E-Notes, vol. 9, no. 2, 2021, pp. 42-52.
Vancouver Porwal S. $q$-Analogue of a New Subclass of Harmonic Univalent Functions Defined by Fractional Calculus Operator. Math. Sci. Appl. E-Notes. 2021;9(2):42-5.

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