Applications of $q$-Srivastava-Attiya operator on subclasses of analytic functions

Year 2025, Volume: 74 Issue: 2, 254 - 266, 19.06.2025

Rizwan Salim Badar

https://doi.org/10.31801/cfsuasmas.1515007

Abstract

The aim of the article is to investigate the applications of $q$-Srivastava-Attiya integral operator on the subclasses of $q$-starlike, $q$-convex and $q$-close-to-convex functions. The results are established by using the subordination result and $q$-analogue of Jack's Lemma. The characteristic properties including necessary condition and its applications, inclusions and integral preservations of the defined classes in the context of $q$-Bernardi operator are established.

Keywords

$q$-calculus , integral operators , $q$-analogue of subclasses of analytic functions , subordination , $q$-Jack lemma

References

• Ademoğulları, K., Kahramaner, Y., Polatoğlu, Y., $q$-harmonic mappings for which analytic part is $q$-convex functions, Nonlinear Anal. Differ. Equ., 4 (2016), 283-293.
• Agrawal, S., Sahoo, S. K., A generalization of starlike functions of order alpha, Hokkaido Math. J., 46 (2017), 15-27.
• Arif, M., Rani, L., Raza, M., Zaprawa, P., Fourth Hankel determinant for the family of functions with bounded turning, Bull. Korean Math. Soc., 55(6) (2018), 1703-1711.
• Arif, M., Srivastava, H. M., Umar, S., Some applications of a $q$-analogue of the Ruscheweyh type operator for multivalent functions, RACSAM, 113 (2019), 1211-1221.
• Arif, M., Ullah, I., Raza, M., Zaprawa, P., Investigation of the fifth Hankel determinant for a family of functions with bounded turnings, Math. Slovaca, 70(2) (2020), 319-328.
• Arif, M., Ul-Haq, M., Liu, J. L., A subfamily of univalent functions associated with $q$-analogue of Noor integral operator, J. Funct. Spac., 3818915 (2018).
• Çetinkaya, A., Polatoğlu, Y., $q$-Harmonic mappings for which analytic part is $q$-convex functions of complex order, Hacet. J. Math. Stat., 47 (2018), 813-820.
• Ernst, T., A Comprehensive Treatment of $q$-Calculus, Springer, 2012.
• Goodman, A. W., Univalent Functions, Polygonal Publishing House, 1983.
• Govindaraj, M., Sivasubramanian, S., On a class of analytic functions related to conic domains involving $q$-calculus, Anal. Mathematica., 43 (2017), 475-487.
• Ismail, M. E. H., Markes, E., Styer, D., A generalization of starlike functions, Complex Var., 14 (1990), 77-84.
• Jackson, F. H., On $q$-functions and certain difference operator, Amer. J. Math., 46 (1908), 253-281.
• Jackson, F. H., On $q$-definite integrals, Q. J. Pure Appl. Math., 41 (1910), 193-203.
• Kanas, S., R˘aducanu, D., Some classes of analytic functions related to conic domains, Math. Slovaca, 64 (2014), 1183-1196.
• Noor, K. I., On generalized $q$-close-to-convexity, Appl. Math. Inf. Sci., 11 (2017), 1383-1388.
• Noor, K. I., Badar, R. S., On a class of quantum alpha-convex functions, J. Appl. Math. Info., 36 (2018), 541-548.
• Noor, K. I., Riaz, S., Generalized $q$-starlike functions, Studia Sci. Math. Hungar., 54 (2017), 509-522.
• Noor, K. I., Riaz, S., Noor, M. A., On $q$-Bernardi integral operator, TWMS J. Pure Appl. Math., 8 (2017), 3-11.
• Khan, Q., Arif, M., Raza, M., Srivastava, G., Tang, H., Rehman, S., Some applications of a new integral operator in $q$-analog for multivalent functions, Mathematics, 7(12) (2019), 1178.
• Răducanu, D., Srivastava, H. M., A new class of analytic functions defined by means of a convolution operator involving the Hurwitz-Lerch Zeta function, Integral Transforms Spec. Funct., 18 (2017), 933-943.
• Sahoo, S. K., Sharma, N. L., On a generalization of close-to-convex functions, Ann. Polon. Math., 113 (2015), 93-108.
• Shah, S. A., Noor, K. I., Study on the $q$-analogue of a certain family of linear operators, Turk. J. Math., 43 (2019), 2707-2714.
• Shamsan, H., Latha, S., On generalized bounded Mocanu variation related to $q$-derivative and conic regions, Ann. Pure Appl. Math., 17 (2018), 67-83.
• Srivastava, H. M., Univalent Functions, Fractional Calculus and Their Applications, John Wiley and Sons, 1989.
• Srivastava, H. M., Operators of basic (or $q$-) calculus and fractional $q$-calculus and their applications in geometric function theory, Iran. J. Sci. Technol. Trans. Sci., 44 (2020), 327-344.
• Srivastava, H. M., Arif, M., Raza, M., Convolution properties of meromorphically harmonic functions defined by a generalized convolution $q$-derivative operator, AIMS Math., 6(6) (2021), 5869-5885.
• Srivastava, H. M., Tahir, M., Khan, B., Ahmed, Q. Z., Khan, N., Some general classes of $q$-starlike functions associated with the Janowski functions, Symmetry, 11(292) (2019).