Family of Analytic Functions with Negative Coefficients Involving $q-$Analogue of Multiplier Transformation Operator

Year 2023, , 138 - 152, 02.09.2023

Tamer Seoudy , Mohamed Kamal Aouf

https://doi.org/10.36753/mathenot.883641

Abstract

We introduce a new class of analytic functions with negative coefficients by using the $q-$analogue of multiplier transformation operator. Coefficient inequalities, distortion theorems, closure theorems, and some properties involving the modified Hadamard products, radii of close-to-convexity, starlikeness, and convexity, and integral operators associated with functions belonging to this class are obtained.

Keywords

Univalent function, convolution, q-convex, q-starlike, q-analogue of multiplier transformation operator

References

• [1] Albayrak, D., Purohit, S. D., Uçar, F.: On q􀀀analogues of Sumudu transforms. Analele Stiintifice ale. Univ. Ovidius Constanta, Ser. Mat. 21 (1), 239-260 (2013).
• [2] Albayrak, D., Purohit, S. D., Uçar, F.: On q􀀀Sumudu transforms of certain q􀀀polynomials. Filomat. 27(2), 411-427 (2013).
• [3] Annaby, M. H., Mansour, Z. S.: q􀀀Fractional calculus and equations. In: Lecture Notes in Mathematics. Springer-Verlag, Berlin Heidelberg (2012).
• [4] Aral, A., Gupta, V., Agarwal, R. P.: Applications of q􀀀calculus in operator theory. Springer. New York (2013).
• [5] Bangerezako, G.: Variational calculus on q􀀀non uniform lattices. J. Math. Anal. Appl. 306(1), 161-179 (2005).
• [6] Kac, V. G., Cheung, P.: Quantum calculus, Universitext. Springer-Verlag. New York (2002).
• [7] Mansour, Z. S. I.: Linear sequential q􀀀difference equations of fractional order. Fract. Calc. Appl. Anal. 12(2), 159-178 (2009).
• [8] Muhyi, A., Araci, S.: A note on q􀀀Fubini-Appell polynomials and related properties. Journal of Function Spaces. 2021, 3805809 (2021).
• [9] Rajkovi´c, P. M., Marinkovi´c, S. D., Stankovi´c, M. S.: Fractional integrals and derivatives in q􀀀calculus. Appl. Anal. Discret. Math. 1, 311-323 (2007).
• [10] Yasmin, G., Muhyi, A.: Certain results of 2􀀀variable q-generalized tangent Apostol type polynomials. J. Math. Comput. Sci. 22(3), 238-251 (2021).
• [11] Aouf, M. K., Darwish, H. E., S˘al˘agean, G. S.: On a generalization of starlike functions with negative coefficients. Mathematica Tome. 43(66)(1), 3-10 (2001).
• [12] Aouf, M. K., Seoudy, T. M.: Convolution properties for classes of bounded analytic functions with complex order defined by q􀀀derivative operator. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 113(2), 1279-1288 (2019).
• [13] Gasper, G., Rahman, M.: Basic hypergeometric series. Cambridge University Press. Cambridge (1990).
• [14] Jackson, F. H.: On q􀀀definite integrals. Quart. J. Pure Appl. Math. 41, 193-203 (1910).
• [15] Seoudy, T. M., Aouf, M. K.: Convolution properties for certain classes of analytic functions defined by q􀀀derivative operator. Abstract and Applied Analysis. 2014, 846719 (2014).
• [16] Seoudy T. M., Aouf, M. K.: Coefficient estimates of new classes of qstarlike and q􀀀convex functions of complex order. J. Math. Inequal. 10(1), 135-145 (2016).
• [17] Jackson, F. H.: On q􀀀functions and a certain difference operator. Transactions of the Royal Society of Edinburgh, 46, 253-281 (1908).
• [18] Ramachandran, C., Vanitha, L., Kanas, S.: Certain results on q􀀀starlike and q􀀀convex error functions. Math. Slovaca. 68(2) 361-368 (2018).
• [19] Govindaraj, M., Sivasubramanian, S.: On a class of analytic functions related to conic domains involving q􀀀calculus., Analysis Mathematica. 43(3), 475-487 (2017).
• [20] S˘al˘agean, G. S.: Subclass of univalent functions. In: Lecture Notes in Mathematics. 1013. Springer-Verlag, 362-372 (1983).
• [21] Aouf, M. K., Cho, N. E.: On a certain subclass of analytic functions with negative cefficients. Turkish Journal of Mathematics. 22(1), 15-32 (1998).
• [22] Aouf, M. K.: A subclass of uniformly convex functions with negative coefficients. Mathematica Tome. 52, 99-111 (2010).
• [23] Aouf, M. K., Seoudy, T. M.: Certain class of bi—Bazilevic functions with bounded boundary rotation involving Salagean operator. Constr. Math. Anal. 3(4), 139-149 (2020).
• [24] Cho, N. E., Kim, T. H.: Multiplier transformations and strongly close-to-convex functions. Bull. Korean Math. Soc. 40, 399-410 (2003).
• [25] Cho, N. E., Srivastava, H. M.: Argument estimates of certain analytic functions defined by a class of multiplier transformations. Math. Comput. Model. 37, 39-49 (2003).
• [26] Uralegaddi B. A., Somanatha, C.: Certain classes of univalent functions. In Current Topics in Analytic Function Theory, (Edited by H.M. Srivastava and S. Owa), World Scientific Publishing Company. Singapore, 371-374 (1992).
• [27] Flett, T. M.: The dual of an identity of Hardy and Littlewood and some related inequalities. J. Math. Anal. Appl. 38, 746-765 (1972).
• [28] Bernardi, S. D.: Convex and starlike univalent functions. Trans. Amer. Math. Soc. 135, 429-446 (1969).
• [29] Aouf, M. K., Srivastava, H. M.: Some families of starlike functions with negative coefficients. J. Math. Anal. Appl. 203, 762-790 (1996).
• [30] Silverman, H.: Univalent functions with negative coefficients. Proc. Amer. Math. Soc. 51, 109-116 (1975). [31] Chatterjea, S. K.: On starlike functions. J. Pure Math. 1, 23-26 (1981).
• [32] Srivastava, H. M., Owa, S., Chatterjea, S. K.: A note on certain classes of starlike functions. Rend. Sem. Mat. Uni. Padova. 77, 115-124 (1987).
• [33] Schild, A., Silverman, H.: Convolutions of univalent functions with negative coefficients. Ann. Uni. Mariae Curie- Skłodowska Sect. A. 29, 99-106 (1975).
• [34] Aldweby H., Darus, M.: A note on q􀀀integral operators. Electronic Notes in Discrete Mathematics. 67, 25-30 (2018).
• [35] Noor, K. I., Riazi, S., Noor, M. A.: On q􀀀Bernardi integral operator. TWMS J. Pure Appl. Math. 8(1), 3-11 (2017).