A novel view of p-valent functions defined by q-derivative and Gegenbauer polynomials

Authors
Ebrahim Amini
Vali Soltani Masih
‎payme Noor University
Abstract
Abstract: In this paper, we define and discuss the properties of various classes of analytic multivalent

functions by using Gegenbauer polynomials. We make use of Gegenbauer polynomials to introduce the

new differential operator. This operator is based on the convolution Hadamard product. Also, we define

a new subclass of p-valent starlike and convex functions of order alpha by using q-derivative operator.

In addition, we derive families of analytic univalent functions associated with Gegenbauer polynomials

and q-derivative operator. Also, we obtain coefficient bounds for the functions in such new subclasses of

analytic functions. Moreover, we obtain several corollaries and results for p-valent starlike and convex

functions of order alpha. We establish certain growth and distortion theorems in such new subclasses

of analytic functions. Moreover, using the concept of the (q, delta)-neighborhood, we provide several

inclusion symmetric relations for certain (q, delta)-neighborhoods of analytic univalent functions with

negative coefficients. Various q-inequities are also discussed in more details
Keywords
Analytic functions
neighbourhood
Hankel determinant
Gegenbauer polynomials
q-derivative
] A. M. Da Cruz, N. Martins, The q-symmetric variational calculus, Comput. Math. Appl., volume 64, (2012), 2241–2250.

[2] A. Lavagno, Basic-deformed quantum mechanics, Rep. Math. Phys., volume 64, ( 2009), 79–88.

[3] F. H. Jackson, q-difference equations, Am. J. Math., volume 32, (1910), 305–314.

[4] F. H. Jackson, On q-functions and a certain difference operator, Trans. R. Soc. Edinb., volume 46(2), 1909, 253-281.

[5] S. D. Purohit, A new class of multivalently analytic functions associated with fractional q-calculus operators, Fract. Differ. Calc., volume 2, (2012), , 129-138.

[6] 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. volume 44, (2020), 327–344.

[7] M. Ismail, E. Merkes, D. Styer, A generalization of starlike functions, Complex Variables, Theory and Application: An International Journal, volume 14 (1990), 77–84.

[8] S. Kanas, S. Altinkaya, and S. Yalçin, Subclass of k-Uniformly Starlike Functions Defined by the Symmetric q-Derivative Operator, Ukr. Math. J., volume 70, (2012), 1727–1740.


[9] X. Zhang, S. Khan, S. Hussain, H. Tang, Z. Shareef, New subclass of q-starlike functions associated with generalized conic domain, AIMS Math., volume 5, (2020), 4830–4848.

[10] S. O. Altinta, H. Irmak, H. Srivastava, Fractional calculus and certain starlike functions with negative coefficients, Comp. Math. Appl., volume 30, (1995), 9–15.

[11] A. Mohammed, M. A. Darus, generalized operator involving the q-hypergeometric function, Mat. Vesn., volume 65, (2013), 454–465.

[12] H. Aldweby,M. Darus, Some subordination results on q-analogue of Ruscheweyh differential operator, Abstr. Appl. Anal., volume 2014, (2014), 958563

[13] H. E. Uçar, Coefficient inequality for q-starlike functions, Appl. Math. Comput., volume 276, ( 2016), 122–126.

[14] M. Arif, H. M. Srivastava, and S. Umar, some applications of a q-analogue of the Ruscheweyh type operator for multivalent functions, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat., volume 113(2), (2019), 1211-1221.

[15] K.S., Nonlaopon, K., and D. Baleanu, Estimates for coefficients of bi-univalent functions associated with a fractional q-difference operator, Symmetry, volume 14(5), (2022), 879.

[16] A. M. C. Brito da Cruz, and N. Martins, The q-symmetric variational calculus, Comput. Math. Appl., volume 64, (2012), 2241–2250.

[17] G. Gasper, M. Rahman, Basic Hypergeometric Series, 2nd ed. Encyclopedia of Mathematics and its Applications, 96. Cambridge University Press, Cambridge, 2004.

[18] I. Graham, G. Kohr, Geometric Function Theory in one and Higher Dimensions, Marcel Dekker, Inc, NewYork, 2003.

[19] J. W. Noonan, D. K. Thomas, On the second Hankel determinant of areally mean p-valent functions, Trans Am Math Soc., volume 223, (1976), 337–346.

[20] S. Khan, S. Hussain, M. Naeem, M. Darus, and A. Rasheed, A subclass of q-starlike functions defined by using a symmetric q-derivative operator and related with generalized symmetric conic domains, Matematiche, volume 9, (2022), 917.


[21] G. Szego, M. Fekete, Eine Bemerkunguber ungerade schlichte Funktionen, J. London Math. Soc., volume 2, (1933), 85–89 .

[22] D. K. Thomas, J. W. Noonan, On the second Hankel determinant of a really mean p-valent functions, Trans. Amer. Math. Soc., volume 223, (1976), 337–346.

[23] R. Vein, P. Dale, Determinants and Their Applicatons im Mathematical Physics, Applied Mathematical Sciences, Springer: New York, NY, USA, 1999; Volume 134.

[24] B. Unyong, V. Govindan S. Bowmiya, Generalized linear differential equation using Hyers–Ulam stability approach, AIMS Math., volume 6(2), (2021), 1607–1623.
[25] S. Khan, A. A. Al-Gonah, G. Yasmin, Generalized and mixed type Gegenbauer polynomials, J. Math Anal Appl. volume 390(1), (2012), 197–207.