Certain subclasses of meromorphically $q$-starlike functions associated with the $q$-derivative operators

H. M. Srivastava; M. Tahir; B.  Khan; M. Darus; N. Khan; Q. Z.  Ahmad

doi:10.37863/umzh.v73i9.814

H. M. Srivastava Univ. Victoria, British Columbia, Canada and China Medical Univ., Taiwan
M. Tahir Abbottabad Univ. Sci. and Technology, Pakistan
B. Khan School Math. Sci. and Shanghai Key Laboratory PMMP, East China Normal Univ., Shanghai, China
M. Darus School Math. Sci., Univ. Kebangsaan Malaysia, Bangi, Selangor, Malaysia
N. Khan Abbottabad Univ. Sci. and Technology, Pakistan
Q. Z. Ahmad Covernment Akhtar Nawaz Khan (Shaheed) Degree College KTS, Haripur, Pakistan

DOI: https://doi.org/10.37863/umzh.v73i9.814

Keywords: Analytic and univalent functions, Meromorphic functions, Meromorphically starlike functions, q-Derivative (or q-difference) operator, Neighborhoods, Partial sums, Ruscheweyh-type q-derivative operator

Abstract

UDC 517.5

The purpose of the present paper is to establish several general results concerning the partial sums of meromorphically starlike functions defined here by means of a certain class of $q$-derivative (or $q$-difference) operators.
The familiar concept of neighborhood for meromorphic functions are also considered.
Moreover, by using a Ruscheweyh-type $q$-derivative operator, we define and study another new class of functions emerging from the class of normalized meromorphic functions.

Author Biographies

H. M. Srivastava, Univ. Victoria, British Columbia, Canada and China Medical Univ., Taiwan

M. Tahir, Abbottabad Univ. Sci. and Technology, Pakistan

M. Darus, School Math. Sci., Univ. Kebangsaan Malaysia, Bangi, Selangor, Malaysia

Q. Z. Ahmad, Covernment Akhtar Nawaz Khan (Shaheed) Degree College KTS, Haripur, Pakistan

References

S. Agrawal, S. K. Sahoo, A generalization of starlike functions of order $alpha$ , Hokkaido Math. J., 46, 15 – 27 (2017), https://doi.org/10.14492/hokmj/1498788094 DOI: https://doi.org/10.14492/hokmj/1498788094

H. Aldweby, M. Darus, Some subordination results on $q$-analogue of Ruscheweyh differential operator, Abstr. and Appl. Anal., 2014, Article ID 958563 (2014), p. 1 – 6, https://doi.org/10.1155/2014/958563 DOI: https://doi.org/10.1155/2014/958563

H. S. Al-Amiri, Certain analogy of the $alpha $-convex functions, Rev. Roumaine Math. Pures Appl., 23, 1449 – 1453 (1978).

M. K. Aouf, H. Silverman, Partial sums of certain meromorphic p-valent functions, J. Inequal. Pure and Appl. Math., 7, №. 4, Article ID 116 (2006), p. 1 – 15.

W. G. Atshan, Subclass of meromorphic functions with positive coefficients defined by Ruscheweyh derivative, II, Surv. Math. and Appl., 3, 67 – 77 (2008).

W. G. Atshan, S. R. Kulkarni, Subclass of meromorphic functions with positive coefficients defined by Ruscheweyh derivative (I), J. Rajasthan Acad. Phys. Sci., 6, 129 – 140 (2007).

A. K. Bakhtin, G. P. Bakhtina, Yu. B. Zelinskii, Topological-algebraic structures and geometric methods in complex analysis (in Russian), Inst. Math. NAS Ukraine, Kyiv (2008).

A. K. Bakhtin, I. V. Denega, Sharp estimates of products of inner radii of non-overlapping domains in the complex plane, Probl. Anal. Issues Anal., 26, №. 1, 17 – 31 (2019). DOI: https://doi.org/10.15393/j3.art.2019.5452

N. E. Cho, S. Owa, Partial sums of certain meromorphic functions, J. Inequal. Pure and Appl. Math., 5, № 2, Article ID 30 (2004), p. 1 – 15.

J. Clunie, On meromorphic schlicht functions, J. London Math. Soc., 34, 215 – 216 (1959), https://doi.org/10.1112/jlms/s1-34.2.215 DOI: https://doi.org/10.1112/jlms/s1-34.2.215

I. V. Denega, Ya. V. Zabolotnii, Estimates of products of inner radii of non-overlapping domains in the complex plane, Complex Var. and Elliptic Equat., 62, № 11, 1611 – 1618 (2017), https://doi.org/10.1080/17476933.2016.1265952 DOI: https://doi.org/10.1080/17476933.2016.1265952

I. V. Denega, Estimates of the inner radii of non-overlapping domains, J. Math. Sci., 242, № 6, 787 – 795 (2019). DOI: https://doi.org/10.1007/s10958-019-04516-2

U. A. Ezeafulukwe, M. Darus, A note on q-calculus, Fasc. Mat., 55, № 1, 53 – 63 (2015). DOI: https://doi.org/10.1515/fascmath-2015-0014

U. A. Ezeafulukwe, M. Darus, Certain properties of q-hypergeometric functions, Int. J. Math. and Math. Sci., 2015, Article ID 489218 (2015), p. 1 – 9, https://doi.org/10.1155/2015/489218 DOI: https://doi.org/10.1155/2015/489218

V. Ya. Gutlyanskii, V. I. Ryazanov, Geometric and topological theory of functions and mappings (in Russian), Naukova Dumka, Kyiv (2011).

A. W. Goodman, Univalent functions and nonanalytic curves, Proc. Amer. Math. Soc., 8, 598 – 601 (1957), https://doi.org/10.2307/2033525 DOI: https://doi.org/10.1090/S0002-9939-1957-0086879-9

M. R. Ganigi, B. A. Uralegaddi, New criteria for meromorphic univalent functions, Bull. Math. Soc. Sci. Math. Roumanie (N. S.), 33, № 81, 9 – 13 (1989).

H. Aldweby, M. Darus, Partial sum of generalized class of meromorphically univalent functions defined by $q$-analogue of Liu – Srivastava operator, Asian-Eur. J. Math., 7, № 03, Article ID 1450046 (2014), p. 1 – 10, https://doi.org/10.1142/S1793557114500466 DOI: https://doi.org/10.1142/S1793557114500466

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

F. H. Jackson, On $q$-definite integrals, Quart. J. Pure and Appl. Math., 41, 193 – 203 (1910).

F. H. Jackson, $q$-Difference equations, Amer. J. Math., 32, 305 – 314 (1910). DOI: https://doi.org/10.2307/2370183

S. Kanas, D. R˘aducanu, Some classes of analytic functions related to conic domains, Math. Slovaca, 64, 1183 – 1196 (2014), https://doi.org/10.2478/s12175-014-0268-9 DOI: https://doi.org/10.2478/s12175-014-0268-9

S. Maharana, J. K. Prajapat, H. M. Srivastava, The radius of convexity of partial sums of convex functions in one direction, Proc. Nat. Acad. Sci. Sect. A, 87, 215 – 219 (2017), https://doi.org/10.1007/s40010-017-0348-7 DOI: https://doi.org/10.1007/s40010-017-0348-7

S. Mahmood, M. Jabeen, S. N. Malik, H. M. Srivastava, R. Manzoor, S. M. J. Riaz, Some coefficient inequalities of $q$-starlike functions associated with conic domain defined by $q$-derivative, J. Funct. Spaces, 2018, Article ID 8492072 (2018), p. 1 – 13, https://doi.org/10.1155/2018/8492072 DOI: https://doi.org/10.1155/2018/8492072

S. Mahmood, Q. Z. Ahmad, H. M. Srivastava, N. Khan, B. Khan, M. Tahir, A certain subclass of meromorphically $q$-starlike functions associated with the Janowski functions, J. Inequal. and Appl., 2019, Article ID 88 (2019), https://doi.org/10.1186/s13660-019-2020-z DOI: https://doi.org/10.1186/s13660-019-2020-z

S. Mahmood, H. M. Srivastava, N. Khan, Q. Z. Ahmad, B. Khan, I. Ali, Upper bound of the third Hankel determinant for a subclass of q-starlike functions, Symmetry, 11, № 3, Article ID 347 (2019), p. 1 – 13. DOI: https://doi.org/10.3390/sym11030347

S. Owa, H. M. Srivastava, N. Saito, Partial sums of certain classes of analytic functions, Internat. J. Comput. Math., 81, 1239 – 1256 (2004), https://doi.org/10.1080/00207160412331284042 DOI: https://doi.org/10.1080/00207160412331284042

S. Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math. Soc., 49, 109 – 115 (1975), https://doi.org/10.2307/2039801 DOI: https://doi.org/10.1090/S0002-9939-1975-0367176-1

S. Ruscheweyh, Neighborhoods of univalent functions, Proc. Amer. Math. Soc., 81, 521 – 527 (1981), https://doi.org/10.2307/2044151 DOI: https://doi.org/10.1090/S0002-9939-1981-0601721-6

H. Silverman, Partial sums of starlike and convex functions, J. Math. Anal. and Appl., 209, 221 – 227 (1997), https://doi.org/10.1006/jmaa.1997.5361 DOI: https://doi.org/10.1006/jmaa.1997.5361

H. M. Srivastava, Univalent functions, fractional calculus, and associated generalized hypergeometric functions, Univalent Functions, Fractional Calculus and Their Applications, John Wiley and Sons, New York ect. (1989), p. 329 – 354.

H. M. Srivastava, 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), https://doi.org/10.1007/s40995-019-00815-0 DOI: https://doi.org/10.1007/s40995-019-00815-0

H. M. Srivastava, ¸ S. Altinkaya, S. Yal¸cın, Hankel determinant for a subclass of bi-univalent functions defined by using a symmetric $q$-derivative operator, Filomat, 32, 503 – 516 (2018), https://doi.org/10.2298/fil1802503s DOI: https://doi.org/10.2298/FIL1802503S

H. M. Srivastava, D. Bansal, Close-to-convexity of a certain family of $q$-Mittag – Leffler functions, J. Nonlinear Var. Anal., 1, 61 – 69 (2017).

H. M. Srivastava, S. Gaboury, F. Ghanim, Partial sums of certain classes of meromorphic functions related to the Hurwitz – Lerch zeta function, Moroccan J. Pure and Appl. Anal., 1, 38 – 50 (2015). DOI: https://doi.org/10.7603/s40956-015-0003-8

H. M. Srivastava, M. Tahir, B. Khan, Q. Z. Ahmad, N. Khan, Some general classes of $q$-starlike functions associated with the Janowski functions, Symmetry, 11, Article ID 292 (2019), p. 1 – 14. DOI: https://doi.org/10.3390/sym11020292

H. M. Srivastava, Q. Z. Ahmad, N. Khan, N. Khan, B. Khan, Hankel and Toeplitz determinants for a subclass of $q$-starlike functions associated with a general conic domain, Mathematics, 7, Article ID 181 (2019), p. 1 – 15. DOI: https://doi.org/10.3390/math7020181

H. M. Srivastava, B. Khan, N. Khan, Q. Z. Ahmad, Coefficient inequalities for q-starlike functions associated with the Janowski functions, Hokkaido Math. J., 48, 407 – 425 (2019), https://doi.org/10.14492/hokmj/1562810517 DOI: https://doi.org/10.14492/hokmj/1562810517