On q-Calculus Related Generalization of Close-to-Convexity

  • Khalida Inayat Noor Department of Mathematics, COMSATS University Islamabad, Islamabad, Pakistan
  • Muhammad Aslam Noor Department of Mathematics, COMSATS University Islamabad, Islamabad, Pakistan
Keywords: convolution, q-calculus, analytic functions, convex and starlike functions, univalent functions, close-to-convexity

Abstract

We introduce and study $q$-analogue of certain classes of analytic functions which are related with generalized close-to-convexity. Distortion, inclusion results and growth rate of coefficient problem are investigated for these classes. Some applications of our results are highlighted.

References

D. A. Brannan, On functions of bounded boundary rotation, Proc. Edin. Math. Soc. 2 (1968/69), 339-347. https://doi.org/10.1017/S001309150001302X

A. W. Goodman, Univalent Functions, Vols. I, II, Polygonal Publishing House, Washington, New Jersey, 1983.

W. K. Hayman, On functions with positive real part, J. Lond. Math. Soc. 36 (1961), 34-48. https://doi.org/10.1112/jlms/s1-36.1.35

M. E. H. Ismail, E. Merkes and D. Styer, A generalization of starlike functions, Complex Variables, Theory Appl. 14 (1990), 77-84. https://doi.org/10.1080/17476939008814407

F. H. Jackson, On q-functions and certain difference operators, Trans. Royal Soc. Edin. 46 (1908), 253-281. https://doi.org/10.1017/S0080456800002751

W. Janowski, Some extremal problems for certain families of analytic functions, Ann. Polon. Math. 28 (1973), 297-326. https://doi.org/10.4064/ap-28-3-297-326

W. Kaplan, Close-to-convex Schlicht functions, Michigan J. 1 (1952), 169-185. https://doi.org/10.1307/mmj/1028988895

M. M. Miller and P. T. Mocanu, Differential Subordinations : Theory and Applications, Marcel Dekker, Inc. New York, Basel, 2000. https://doi.org/10.1201/9781482289817

K. I. Noor, On a generalization of close-to-convexity, Int. J. Math. Math. Sci. 6 (1983), 327-334. https://doi.org/10.1155/S0161171283000289

K. I. Noor, On Hankel determinant for close-to-convex functions, Int. J. Math. Math. Sci. 3 (1980), 477-481. https://doi.org/10.1155/S016117128000035X

K. I. Noor, M. A. Noor and H. M. Mohamed, Quantum approach to starlike functions, Appl. Math. Inform. Sci. 15(4) (2021), 437-441. https://doi.org/10.18576/amis/150405

K. I. Noor and M. A. Noor, Hankel determinant problem for q-strongly close-to-convex functions, Earthline J. Math. Sci. 8(2) (2022), 227-236. https://doi.org/10.34198/ejms.8222.227236

K. I. Noor, On generalized q-close-to-convexity, Appl. Math. Inform. Sci. 11(5) (2017), 1383-1388. https://doi.org/10.18576/amis/110515

K. I. Noor, On anlytic functions involving q-Ruscheweyh derivative, Fractal and Fractional 3(1) (2019), 1-8. https://doi.org/10.3390/fractalfract3010010

K. I. Noor and S. Riaz, Generalized q-strarlike functions, Studia Sci. Math. Hungarica 54(4) (2017), 509-522. https://doi.org/10.1556/012.2017.54.4.1380

K. I. Noor, S. Riaz and M. A. Noor, On q-Bernardi integral operator, TWMS J. Pure Appl. Math. 8(1) (2017), 3-11.

K. I. Noor, S. Altinkay and S. Yalcin, Coefficient inequalities of analytic functions equipped with conic domains involving q-analogue of Noor integral operator, Tibilisi Math. J. 14(1) (2021), 1-14. https://doi.org/10.32513/tmj/1932200811

K. I. Noor and S. A. Shah, On q-Mocanu type functions associated with q-Rusheweyh derivative operator, Inter. J. Anal. Appl. 18(4) (2020), 550-558.

K. I. Noor and M. A. Noor, On q-analytic functions involving Carlson-Shaffer Operator, J. Advan. Math. Stud. 15(3) (2022), 243-249.

K. Piejko and J. Sokół, On convolution and q-calculus, Bol. Soc. Mat. Mex. 26 (2020), 349-359. https://doi.org/10.1007/s40590-019-00258-y

S. Ruscheweyh and T. Shiel-Small, Hadamard product of Schlicht functions and the Polya-Schengerg conjecture, Comment. Math. Helv. 48 (1973), 119-135. https://doi.org/10.1007/BF02566116

H. E. O. Ucar, Coefficient inequalitis for q-starlike functions, Appl. Math. Comput. 276 (2016), 122-126. https://doi.org/10.1016/j.amc.2015.12.008

Published
2023-03-24
How to Cite
Noor, K. I., & Noor, M. A. (2023). On q-Calculus Related Generalization of Close-to-Convexity. Earthline Journal of Mathematical Sciences, 12(2), 165-182. https://doi.org/10.34198/ejms.12223.165182
Section
Articles

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