New subclasses of bi-univalent functions connected with a q-analogue of convolution based upon the Legendre polynomials

Sheza M. El-Deeb, Bassant M. El-Matary

Abstract


In this paper, we introduce new subclasses of analytic and bi-univalent functions connected with a q-analogue of convolution by using the Legendre  polynomials. Furthermore, we find estimates on the first two Taylor-Maclaurin coefficients for functions in these subclasses and obtain Fekete-Szegő problem for these subclasses.

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Abu Risha, M.H., Annaby, M.H., Ismail, M.E.H., Mansour, Z.S.,

{it Linear $q$-difference equations}, Z. Anal. Anwend., {bf 26}(2007), 481-494.

Arif, M., Ul Haq, M., Liu, J.L., {it A subfamily of univalent functions associated with $q$-analogue of Noor integral operator}, J. Function Spaces, (2018), Art. ID 3818915, 1-5, https://doi.org/10.1155/2018/3818915.

Brannan, D.A., Clunie, J., Kirwan, W.E., {it Coefficient estimates for a class of starlike functions}, Canad. J. Math., {bf 22}(3)(1970), 476-485.

Brannan, D.A., Taha, T.S., {it On some classes of bi-univalent functions}, in: S. M. Mazhar, A. Hamoui, N. S. Faour (Eds.), Mathematical

Analysis and its Applications, Kuwait; February 18-21, 1985, in: KFAS

Proceedings Series, vol. 3, Pergamon Press (Elsevier Science Limited),

Oxford, 1988, pp. 53-60; see also Studia Univ. Babec{s}-Bolyai Math.,

(2)(1986), 70-77.

Bulboacu{a}, T., {it Differential Subordinations and Superordinations, Recent Results}, House of Scientific Book Publ., Cluj-Napoca, 2005.

Duren, P.L., {it Univalent Functions}, Grundlehren der Mathematischen Wissenschaften, {bf 259}, Springer-Verlag, New York, Berlin, Heidelberg and Tokyo, 1983.

El-Deeb, S.M., {it Maclaurin coefficient estimates for new

subclasses of bi-univalent functions connected with a q-analogue of Bessel function}, Abstract Appl. Analy., (2020), Article ID 8368951, 1-7,

https://doi.org/10.1155/2020/8368951.

El-Deeb, S.M., Bulboacu{a}, T., {it Fekete-SzegH{o} inequalities

for certain class of analytic functions connected with $q$-anlogue of Bessel function}, J. Egyptian. Math. Soc., (2019), 1-11,

https://doi.org/10.1186/s42787-019-0049-2.

El-Deeb, S.M., Bulboacu{a}, T., {it Differential sandwich-type

results for symmetric functions connected with a $q$-analog integral

operator}, Mathematics, {bf 7}(2019), no. 12, 1-17, https://doi.org/10.3390/math7121185.

El-Deeb, S.M., Bulboacu{a}, T., {it Differential sandwich-type

results for symmetric functions associated with Pascal distribution series},

J. Contemporary Math. Anal. (in press).

El-Deeb, S.M., Bulboacu{a}, T., Dziok, J., {it Pascal distribution series connected with certain subclasses of univalent functions},

Kyungpook Math. J., {bf 59}(2019), 301-314.

El-Deeb, S.M., Bulboacu{a}, T., El-Matary, B.M.,

{it Maclaurin coefficient estimates of bi-univalent functions connected with the q-derivative}, Mathematics, {bf 8}(2020), 1-14,

https://doi.org/10.3390/math8030418.

Gasper, G., Rahman, M., {it Basic Hypergeometric Series (with a Foreword by Richard Askey)}, Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, {bf 35}, 1990.

Jackson, F.H., {it On $q$-functions and a certain difference

operator}, Trans. Royal Soc. Edinburgh, {bf 46}(1909), no. 2, 253-281,

https://doi.org/10.1017/S0080456800002751

Jackson, F.H., {it On $q$-definite integrals},

Quart. J. Pure Appl. Math., {bf 41}(1910), 193-203.

Lebedev, N., {it Special Functions and Their Applications},

Dover, New York, 1972.

Miller, S.S., Mocanu, P.T., {it Differential Subordinations. Theory and Applications}, Series on Monographs and Textbooks in Pure and

Applied Mathematics, Vol. {bf 225}, Marcel Dekker Inc., New York and Basel, 2000.

Nehari, Z., {it Conformal Mapping}, McGraw-Hill, New York, NY, 1952.

Porwal, S., {it An application of a Poisson distribution series on

certain analytic functions}, J. Complex Anal., (2014), Art. ID 984135, 1-3, https://dx.doi.org/10.1155/2014/984135.

Prajapat, J.K., {it Subordination and superordination preserving

properties for generalized multiplier transformation operator},

Math. Comput. Modelling, {bf 55}(2012), 1456-1465.

Srivastava, H.M., {it Certain $q$-polynomial expansions for functions of several variables}, I and II, IMA J. Appl. Math., {bf 30}(1983),

-209.

Srivastava, H.M., {it Univalent functions, fractional calculus,

and associated generalized hypergeometric functions},

in Univalent Functions, Fractional Calculus, and Their Applications (H.M. Srivastava and S. Owa, Editors), Halsted Press (Ellis Horwood Limited, Chichester), 329-354, John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1989.

Srivastava, H.M., {it 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. Sci., {bf 44}(2020), 327-344.

Srivastava, H.M., El-Deeb, S.M., {it A certain class of

analytic functions of complex order with a $q$-analogue of integral

operators}, Miskolc Math. Notes, {bf 21}(2020), no. 1, 417-433.

Srivastava, H.M., El-Deeb, S.M., {it The Faber polynomial

expansion method and the Taylor-Maclaurin coefficient estimates of

bi-close-to-convex functions connected with the q-convolution},

AIMS Math., {bf 5}(6)(2020), 7087-7106.

Srivastava, H.M., Karlsson, P.W., {it Multiple Gaussian Hypergeometric Series}, Wiley, New York, 1985.

Srivastava, H.M., Khan, S., Ahmad, Q.Z., Khan, N., Hussain, S.,

{it The Faber polynomial expansion method and its application to the

general coefficient problem for some subclasses of bi-univalent functions

associated with a certain $q$-integral operator}, Stud. Univ. Babec{s}-Bolyai Math., {bf 63}(2018), 419-436. 28. Srivastava, H.M., Mishra, A.K., Gochhayat, P., {it Certain subclasses of analytic and bi-univalent functions}, Appl. Math. Lett., {bf 23}(10)(2010), 1188-1192.




DOI: http://dx.doi.org/10.24193/subbmath.2023.3.06

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