×
Wongsaijai, Ben; Sukantamala, Nattakorn

Applications of fractional \(q\)-calculus to certain subclass of analytic \(p\)-valent functions with negative coefficients. (English) Zbl 1354.30012

Abstr. Appl. Anal. 2015, Article ID 273236, 12 p. (2015).
Summary: By making use of the concept of fractional \(q\)-calculus, we firstly define \(q\)-extension of the generalization of the generalized Al-Oboudi differential operator. Then, we introduce new class of \(q\)-analogue of \(p\)-valently closed-to-convex function, and, consequently, new class by means of this new general differential operator. Our main purpose is to determine the general properties on such class and geometric properties for functions belonging to this class with negative coefficient. Further, the \(q\)-extension of interesting properties, such as distortion inequalities, inclusion relations, extreme points, radii of generalized starlikeness, convexity and close-to-convexity, quasi-Hadamard properties, and invariant properties, is obtained. Finally, we briefly indicate the relevant connections of our presented results to the former results.

Cited in 6 Documents

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
26A33 Fractional derivatives and integrals

Keywords:

new classes of analytic functions in the disk; star-like functions; convex functions
PDF BibTeX XML Cite
\textit{B. Wongsaijai} and \textit{N. Sukantamala}, Abstr. Appl. Anal. 2015, Article ID 273236, 12 p. (2015; Zbl 1354.30012)
Full Text: DOI

References:

[1] Ismail, M. E.; Merkes, E.; Styer, D., A generalization of starlike functions, Complex Variables, Theory and Application, 14, 1-4, 77-84 (1990) · Zbl 0708.30014
[2] Agrawal, S.; Sahoo, S. K., On a generalization of close-to-convex functions, Annales Polonici Mathematici, 113, 1, 93-108 (2015) · Zbl 1308.30020
[3] Raghavendar, K.; Swaminathan, A., Close-to-convexity of basic hypergeometric functions using their Taylor coefficients, Journal of Mathematics and Applications, 35, 111-125 (2012) · Zbl 1382.30035
[4] Al Dweby, H.; Darus, M., On harmonic meromorphic functions associated with basic hypergeometric functions, The Scientific World Journal, 2013 (2013)
[5] Aldweby, H.; Darus, M., Some subordination results on \(q\)-analogue of Ruscheweyh differential operator, Abstract and Applied Analysis, 2014 (2014) · Zbl 1474.30044
[6] Murugusundaramoorthy, G.; Selvaraj, C.; Babu, O. S., Subclasses of starlike functions associated with fractional \(q\)-calculus operators, Journal of Complex Analysis, 2013 (2013) · Zbl 1268.30024
[7] Selvakumaran, K. A.; Purohit, S. D.; Secer, A., Majorization for a class of analytic functions defined by \(q\) -differentiation, Mathematical Problems in Engineering, 2014 (2014) · Zbl 1407.30004
[8] Selvakumaran, K. A.; Purohit, S. D.; Secer, A.; Bayram, M., Convexity of certain \(q\)-integral operators of \(p\)-valent functions, Abstract and Applied Analysis, 2014 (2014) · Zbl 1474.30116
[9] Salagean, G. S., Subclasses of univalent functions, Complex Analysis—Fifth Romanian-Finnish Seminar. Complex Analysis—Fifth Romanian-Finnish Seminar, Lecture Notes in Mathematics, 1013, 362-372 (1983), Berlin, Germany: Springer, Berlin, Germany
[10] Al-Shaqsi, K.; Darus, M.; Fadipe-Joseph, O. A., A new subclass of Salagean-type harmonic univalent functions, Abstract and Applied Analysis, 2010 (2010) · Zbl 1207.30010
[11] El-Ashwah, R. M.; Aouf, M. K.; Hassan, A. A.; Hassan, A. H., A new class of analytic functions defined by using Salagean operator, International Journal of Analysis, 2013 (2013) · Zbl 1268.30012
[12] Aouf, M. K., Subordination properties for a certain class of analytic functions defined by the Salagean operator, Applied Mathematics Letters, 22, 10, 1581-1585 (2009) · Zbl 1171.30302
[13] Goswami, P.; Aouf, M. K., Majorization properties for certain classes of analytic functions using the Salagean operator, Applied Mathematics Letters, 23, 11, 1351-1354 (2010) · Zbl 1201.30014
[14] Aouf, M. K.; Seoudy, T. M., On differential Sandwich theorems of analytic functions defined by generalized Sălăgean integral operator, Applied Mathematics Letters, 24, 8, 1364-1368 (2011) · Zbl 1216.30002
[15] Al-Oboudi, F. M., On univalent functions defined by a generalized Sălăgean operator, International Journal of Mathematics and Mathematical Sciences, 2004, 27, 1429-1436 (2004) · Zbl 1072.30009
[16] Al-Oboudi, F. M.; Al-Amoudi, K. A., On classes of analytic functions related to conic domains, Journal of Mathematical Analysis and Applications, 339, 1, 655-667 (2008) · Zbl 1132.30010
[17] Bulut, S., The generalization of the generalized Al-Oboudi differential operator, Applied Mathematics and Computation, 215, 4, 1448-1455 (2009) · Zbl 1176.30027
[18] Vijaya, R.; Sudharsan, T. B.; Murugusundaramoorthy, G., A class of analytic functions based on an extension of generalized Salagean operator, Engineering Mathematics Letters, 2014, article 16 (2014)
[19] Aouf, M. K.; Mostafa, A. O., Some subordination results for classes of analytic functions defined by the Al-Obouid-AlAmoudi operator, Archiv der Mathematik, 92, 3, 279-286 (2009) · Zbl 1165.30313
[20] Bulut, S., Coefficient inequalities for certain subclasses of analytic functions defined by using a general derivative operator, Kyungpook Mathematical Journal, 51, 3, 241-250 (2011) · Zbl 1230.30008
[21] Eker, S. S.; Güney, H., A new subclass of analytic functions involving Al-Oboudi differential operator, Journal of Inequalities and Applications, 2008 (2008) · Zbl 1151.30009
[22] Kadioglu, E., On subclass of univalent functions with negative coefficients, Applied Mathematics and Computation, 146, 2-3, 351-358 (2003) · Zbl 1030.30008
[23] Al-Oboudi, F. M.; Al-Amoudi, K. A., Subordination results for classes of analytic functions related to conic domains defined by a fractional operator, Journal of Mathematical Analysis and Applications, 354, 2, 412-420 (2009) · Zbl 1165.30307
[24] Goodman, A. W., Univalent Functions, Vols. I and II (1983), Tampa, Fla, USA: Mariner Publishing, Tampa, Fla, USA
[25] Umezawa, T., On the theory of univalent functions, Tohoku Mathematical Journal, 2, 7, 212-228 (1955) · Zbl 0066.32603
[26] Umezawa, T., Multivalently close-to-convex functions, Proceedings of the American Mathematical Society, 8, 869-874 (1957) · Zbl 0080.28301
[27] Gasper, G.; Rahman, M., Basic Hypergeometric Series (1990), Cambridge, UK: Cambridge University Press, Cambridge, UK · Zbl 0695.33001
[28] Jackson, F. H., On \(q\)-functions and a certain difference operator, Transactions of the Royal Society of Edinburgh, 46, 64-72 (1908)
[29] Purohit, S. D.; Raina, R. K., Certain subclasses of analytic functions associated with fractional \(q\)-calculus operators, Mathematica Scandinavica, 109, 1, 55-70 (2011) · Zbl 1229.33027
[30] Owa, S.; Srivastava, H. M., Univalent and starlike generalized hypergeometric functions, Canadian Journal of Mathematics, 39, 5, 1057-1077 (1987) · Zbl 0611.33007
[31] Acu, M.; Owa, S., Note on a class of starlike functions, Proceedings of the RIMS
[32] Acu, M.; Polataglu, Y.; Yavuz, E., The radius of starlikeness of the certain classes of p-valent functions defined by multiplier transformations, Journal of Inequalities and Applications, 2008 (2008) · Zbl 1151.30308
[33] Cătaş, A., Sandwich theorems associated with new multiplier transformations, Journal of Computational Analysis and Applications, 13, 4 (2011) · Zbl 1218.30020
[34] Cho, N. E.; Srivastava, H. M., Argument estimates of certain analytic functions defined by a class of multiplier transformations, Mathematical and Computer Modelling, 37, 1-2, 39-49 (2003) · Zbl 1050.30007
[35] Cho, N. E.; Kim, T. H., Multiplier transformations and strongly close-to-convex functions, Bulletin of the Korean Mathematical Society, 40, 3, 399-410 (2003) · Zbl 1032.30007
[36] Kumar, S. S.; Taneja, H. C.; Ravichandran, V., Classes of multivalent functions defined by Dziok-Srivastava linear operator and multiplier transformation, Kyungpook Mathematical Journal, 46, 1, 97-109 (2006) · Zbl 1104.30016
[37] Uralegaddi, B. A.; Somanatha, C., Certain classes of univalent functions, Current Topics in Analytic Function Theory, 371-374 (1992), River Edge, NJ, USA: World Scientific Publishers, River Edge, NJ, USA · Zbl 0987.30508
[38] Bulut, S., On a class of analytic functions defined by generalized Al-Oboudi differential operator, Tamsui Oxford Journal of Information and Mathematical Sciences, 28, 3, 292-301 (2012) · Zbl 1307.30019
[39] Tăut, A. O.; Oros, G. I.; Şendruţiu, R., On a class of univalent functions defined by Salagean differential operator, Banach Journal of Mathematical Analysis, 3, 1, 61-67 (2009) · Zbl 1155.30339
[40] Altintas, O., A subclass of analytic functions with negative coefficients, Hacettepe Bulletin of Natural Sciences and Engineering, 19, 15-24 (1990) · Zbl 0759.30006
[41] Schild, A.; Silverman, H., Convolution of univalent functions with negative coefficients, Annales Universitatis Mariae Curie—Skłodowska Sectio A: Mathematica, 29, 99-107 (1975) · Zbl 0363.30018
[42] Bernardi, S. D., Convex and starlike univalent functions, Transactions of the American Mathematical Society, 135, 429-446 (1969) · Zbl 0172.09703
[43] Libera, R. J., Some classes of regular functions, Proceedings of the American Mathematical Society, 16, 755-758 (1965) · Zbl 0158.07702
[44] Livingston, A. E., On the radius of univalence of certain analytic functions, Proceedings of the American Mathematical Society, 17, 352-357 (1966) · Zbl 0158.07701
[45] Sarangi, S. M.; Uralegaddi, B. A., The radius of convexity and starlikeness for certain classes of analytic functions with negative coefficients I, Rendiconti Accademia Nazionale dei Lincei, 65, 38-42 (1978) · Zbl 0433.30008
[46] Aouf, M. K.; Hossen, H. M.; Srivastava, H. M., A certain subclass of analytic \(p\)-valent functions with negative coefficients, Demonstratio Mathematica, 31, 3, 595-608 (1998) · Zbl 0928.30009
[47] Aouf, M. K., The quasi-Hadamard products of certain subclasses of analytic \(p\)-valent functions with negative coefficients, Applied Mathematics and Computation, 187, 1, 54-61 (2007) · Zbl 1115.30009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.