Applications of Higher-Order Derivatives to the Subclasses of ‎Meromorphic Starlike Functions

Document Type : Research Paper


1 School of Mathematical Sciences, East China Normal University, 500 Dongchuan Road, Shanghai 200241, People’s Republic of China‎

2 Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada

3 Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan, Republic of China

4 Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, AZ1007 Baku, Azerbaijan

5 Department of Mathematics, Abbottabad University of Science and Technology, Abbottabad 22010, Pakistan‎

6 Department of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia

7 School of mathematics Northwest University Xi'an 710127 P. R. China


In this paper, we introduce and study some new classes of multivalent (p -valent) meromorphically starlike functions involving Higher-Order derivatives. For these multivalent classes of functions, we derive several interesting properties including sharp coefficient bounds, neighborhoods, partial sums and inclusion relationships. For validity of our results relevant connections with those in earlier works are also pointed out.


Main Subjects

[1] Kanas, S., Wisniowska, A., Conic regions and k-uniform convexity, J. Comput. Appl. Math., 105, 1999, 327-336.
[2] Kanas, S., Coefficient estimates in subclasses of the Caratheodory class related to conical domains, Acta Math. Univ. Comenian., 74, 2005, 149-161.
[3] Kanas S., Srivastava, H.M., Linear operators associated with k-uniformly convex functions, Integral Transforms and Special Functions,9, 2000, 121-132.
[4] Khan, N., Shafiq, M., Darus, M., Khan, B., Ahmad, Q.Z., Upper bound of the third Hankel determinant for a subclass of q-starlike functions associated with lemniscate of Bernoulli, J. Math. Inequal., 14(1), 2020, 51--63.
[5] Aouf M.K., Mostafa, A.O., On partial sum of certain meromorphic p-valent functions, Math. Comput. Model., 50, 2009, 1325-1331.
[6] Aouf, M.K., Hossen, H.M., New criteria for meromorphic p-valent starlike functions, Tsukuba J. Math.,50(17), 1993, 481-486.
[7] Ali, R.M., Ravichanran, V., Classes of meromorphic a-convex functions, Taiwan. J. Math., 14, 2010, 1479-1490.
[8] Ali, R.M., Seenivasagan, N., Subordination and superordination of the Liu-Srivastava linear operator on meromorphic functions, Bull. Malays. Math. Soc., 31, 2008, 193-207.
[9] Albehbah, M., Darus, M., New subclass of multivalent hypergeometric meromorphic functions, Kragujevac J. Math., 42(1), 2018, 83-95.
[10] Darus, M., Akbarally, A., Coefficient estimates for Ruscheweyh derivative, Int. J. Math. Math. Sci., 2004, 2004, 1937-1942.
[11] Liu, J.-L., Srivastava, H.M., Some convolution conditions for starlikeness and convexity of meromorphically multivalent functions, Appl. Math. Lett., 16, 2003, 13-16.
[12] Liu, J.-L., Srivastava, H.M., A linear operator and associated families of mermorhically multivalent fuctions, J. Math. Anal. Appl., 259, 2001, 566-581.
[13] Liu, J.-L., Srivastava, H.M., Classes of meromorphically multivalent functions associated with the generalized hypergeometric function, Math. Comput. Model., 39, 2004, 35-44.
[14] Liu, J.-L., Srivastava, H.M., Subclasses of meromorphically multivalent functions associated with a certain linear operator, Math. Comput., 193, 2007, 1-6.
[15] Prasin, B.A., Generalization of partial sum of certain analytic and univalent functions, Appl. Math. Lett.,21, 2008, 735-741.
[16] Mayilvaganan, S., Magesh, N., Gatti, N.B., On certain subclasses of meromorphic functions with positive coefficients associated with Liu-Srivastava linear operator, J. Pure. Appl. Math., 113, 2017, 132–141.
[17] Mahmood, S., Ahmad, Q.Z., Srivastava, H.M., Khan, N., Khan, B., Tahir, M., A certain subclass of meromorphically q-starlike functions associated with the Janowski functions, J. Inequal. Appl., 2019, 2019, Art. 88.
[18] Mostafa, A.O., Aouf, M.K., Zayed, H.M., Bulboacă, T., Convolution conditions for subclasses of meromorphic functions of complex order associated with basic Bessel functions, J. Egypt. Math. Soc., 25, 2017, 286-290.
[19] Rehman, M.S., Ahmad, Q.Z., Srivastava, H.M., Khan, B., Khan, N., Partial sums of generalized q-Mittag-Leffler functions, AIMS Math., 5(1), 2019, 408-420.
[20] Khan, M.N., Ahmad, I., Ahmad, H., A radial basis function collocation method for space-dependent inverse heat problems, J. Appl. Comput. Mech., 2020, DOI: 10.22055/JACM.2020.32999.2123.
[21] Ahmad, I., Ihsan, M., Hussain, I., Kumam, P., Kumam, W., Numerical simulation of PDEs by local meshless differential quadrature collocation method, Symmetry, 11(3), 2019, 394.
[22] Ahmad, I., Khan, M.N., Mustafa Inc., Ahmad, H., Nisar, K.S., Numerical simulation of simulate an anomalous solute transport model via local meshless method, Alexandria Engineering Journal, 59(4), 2020, 2827-2838.
[23] Srivastava, H.M., Ahmad, H., Ahmad, I., Thounthong, P., Khan, M.N., Numerical simulation of three-dimensional fractional-order convection-diffusion PDEs by a local meshless method, Thermal Science, 2020, 1-14, DOI: 10.2298/TSCI200225210S.
[24] Ahamd, I., Ahmad, H., Thounthong, P., Chu, Y.M., Cesarano, C., Solution of multi-term time-fractional PDE models arising in mathematical biology and physics by local meshless method, Symmetry, 12(7), 2020, 1195.
[25] Khan, M.N., Siraj-ul-Islam, Hussain, I., Ahmad, H., A local meshless method for the numerical solution of space dependent inverse heat problems, Mathematical Methods in the Applied Sciences, 2020, DOI: 10.1002/mma.6439.
[26] Ahmad, I., Siraj-ul-Islam, Mehnaz, Zaman, S., Local meshless differential quadrature method for time fractional PDEs, AIMS Math., 13(10), 2020, 2641-2654.
[27] Ahmad, I., Riaz, M., Ayaz, M., Arif, M., Siraj, I., Kumam, P., Numerical simulation of partial differential equations via local meshless method, Symmetry, 11(2), 2019, 257.
[28] Ahmad, I., Ahsan, M., Zaheer-ud-Din, Ahmad, M., Kumam, P., An efficient local foundation for time dependent PDEs, Mathematics, 7(3), 2019, 216.
[29] Abo‐Dahab, S.M., Abouelregal, A.E., Ahmad, H., Fractional heat conduction model with phase lags for a half‐space with thermal conductivity and temperature dependent, Mathematical Methods in the Applied Sciences, 2020, DOI: 10.1002/mma.6614.
[31] Ahmad, H., Khan, T.A., Variational iteration algorithm I with an auxiliary parameter for the solution of differential equations of motion for simple and damped mass–spring systems, Noise & Vibration Worldwide., 51(1-2), 2020, 12-20.
[32] Khan, B., Liu, Z.-G., Srivastava, H.M., Khan, N., Darus, M., Tahir, M., A study of some families of multivalent q-starlike functions involving higher-order q-derivatives, Mathematics, 8, 2020, 1470.
[33] Khan, B., Srivastava, H.M., Khan, N., Darus, M., Tahir, M., Ahmad, Q.Z., Coefficient estimates for a subclass of analytic functions associated with a certain leaf-like domain, Mathematics, 8, 2020, 1334.
[34] Rehman, M.S., Ahmad, Q.Z., Khan, B., Tahir, M., Khan, N., Generalisation of certain subclasses of analytic and univalent functions, Maejo Internat. J. Sci. Technol., 13(1), 2019, 1-9.
[35] Ahmad, Q.Z., Khan, N., Raza, N., Tahir, M., Khan, B., Certain q-difference operators and their applications to the subclass of meromorphic q-starlike functions, Filomat, 33(11), 2019, 3385-3397.
[36] Shafiq, M., Khan, N., Srivastava, H.M., Khan, B., Ahmad, Q.Z., Tahir, M., Generalisation of close-to-convex functions associated with Janowski functions, Maejo Internat. J. Sci. Technol., 14, 2020, 141-155.
[37] Srivastava, H.M., Tahir, M., Khan, B., Ahmad, Q.Z., Khan, N., Some general families of q-starlike functions associated with the Janowski functions, Filomat, 33(9), 2019, 2613-2626.
[38] Wang, Z.-G., Sun, S.-X., Shi, L., Neighborhoods and partial sums for certain subclasses of starlike functions, J. Ineqaul. Appl., 25, 2013, 163.
[39] Wang, Z.-G., Srivastava, H.M., Yuan, S.-M., Some basic properties of certain sublclasses of meromorphically starlike functions, J. Inequal. Appl.,2014, 2014, 29.
[40] Wang, Z.-G., Liu, Z.-H., Xiang, R.-G., Some criteria for meromorphic multivalent starlike functions, Appl. Math. Comput., 218, 2011, 1107-1111.
[41] Goodman, A.W., Univalent Functions, Vols. I, II, Polygonal Publishing House, Washington, New Jersey, 1983.
[42] Dziok, J., Classes of meromorphic functions associated with conic regions, Acta Math. Sci.,32, 2012, 765-774.
[43] Goodman, A.W., Univalent functions and nonanalytic curves, Proc. Amer. Math. Soc., 8, 1957, 598-601.
[44] Ruscheweyh, S., Neighbourhoods of univalent functions, Proc. Amer. Math. Soc., 81, 1981, 521-527.
[45] 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, 2020, 327-344. 
[46] 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, 2020, Article ID 629, 1-16.
[47] Srivastava, H.M., Khan, B., Khan, N., Ahmad, Q.Z., Coefficient inequalities for q-starlike functions associated with the Janowski functions, Hokkaido Math. J., 48, 2019, 407-425.
[48] Srivastava, H.M., Ahmad, Q.Z., Khan, N., Khan, N., Khan, B.,  Hankel and Toeplitz determinants for a subclass of q-starlike functions associated with a general conic domain, Mathematics, 7(2), 2019, 181, 1-15.