Certain properties of \(q\)-hypergeometric functions. (English) Zbl 1476.30044
Summary: The quotients of certain \(q\)-hypergeometric functions are presented as \(g\)-fractions which converge uniformly in the unit disc. These results lead to the existence of certain \(q\)-hypergeometric functions in the class of either \(q\)-convex functions, \(\mathcal{P} \mathcal{C}_q\), or \(q\)-starlike functions \(\mathcal{P} \mathcal{S}_q^{*}\).
MSC:
| 30C45 | Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) |
| 33C05 | Classical hypergeometric functions, \({}_2F_1\) |
References:
| [1] | Jackson, F. H., On \(q\)-functions and a certain difference operator, Transactions of the Royal Society of Edinburgh, 46, 02, 253-281 (1909) · doi:10.1017/s0080456800002751 |
| [2] | Jackson, F. H., \(q\)-Difference equations, The American Journal of Mathematics, 32, 4, 305-314 (1910) · JFM 41.0502.01 · doi:10.2307/2370183 |
| [3] | Sofonea, D. F., Some new properties in \(q\)-calculus, General Mathematics, 16, 1, 47-54 (2008) · Zbl 1240.05022 |
| [4] | Srivastava, H. M.; Owa, S., Univalent Functions, Fractional Calculus, and Their Applications (1989), New York, NY, USA: Halsted Press, Ellis Horwood Limited, Chichester, UK; John Wiley & Sons, New York, NY, USA · Zbl 0683.00012 |
| [5] | Ismail, M. E. H.; Merkes, E.; Styer, D., A generalization of stalike functions, Complex Variables, Theory and Application, 14, 77-84 (1990) · Zbl 0708.30014 |
| [6] | Agrawal, S.; Sahoo, S. K., A generalization of starlike functions of order alpha · Zbl 1361.30017 |
| [7] | Sahoo, S. K.; Sharma, N. L., On a generalization of close-to-convex functions, Annales Polonici Mathematici, 113, 17, 93-108 (2015) · Zbl 1308.30020 |
| [8] | 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 |
| [9] | Kanas, S.; Răducanu, D., Some class of analytic functions related to conic domains, Mathematica Slovaca, 64, 5, 1183-1196 (2014) · Zbl 1349.30054 · doi:10.2478/s12175-014-0268-9 |
| [10] | Murugusundaramoorthy, G.; Janani, T.; Darus, M., Coefficient estimate of biunivalent functions based on \(q\)-hypergeometric functions, Applied Sciences, 17, 75-85 (2015) · Zbl 1325.30014 |
| [11] | Aldweby, H.; Darus, M., A subclass of harmonic univalent functions associated with \(q\)-analogue of Dziok-Srivatava operator, ISRN Mathematical Analysis, 2013 (2013) · Zbl 1286.30002 · doi:10.1155/2013/382312 |
| [12] | Aldweby, H.; Darus, M., On harmonic meromorphic functions associated with basic hypergeometric functions, The Scientific World Journal, 2013 (2013) · doi:10.1155/2013/164287 |
| [13] | Aldweby, H.; Darus, M., Partial sum of generalized class of meromorphically univalent functions defined by \(q\)-analogue of Liu-Srivastava operator, Asian-European Journal of Mathematics, 7, 3 (2014) · Zbl 1300.30016 · doi:10.1142/S1793557114500466 |
| [14] | Baricz, Á.; Swaminathan, A., Mapping properties of basic hypergeometric functions, Journal of Classical Analysis, 5, 2, 115-128 (2014) · Zbl 1412.33027 |
| [15] | Aldweby, H.; Darus, M., Properties of a subclass of analytic functions defined by generalized operator involving \(q\)-hypergeometric function, Far East Journal of Mathematical Sciences, 81, 2, 189-200 (2013) · Zbl 1291.30055 |
| [16] | Aldweby, H.; Darus, M., Some subordination results on \(q\)-analogue of Ruscheweyh differential operator, Abstract and Applied Analysis, 2014 (2014) · Zbl 1474.30044 · doi:10.1155/2014/958563 |
| [17] | Mohammed, A.; Darus, M., A generalized operator involving the q-hypergeometric function, Matematicki Vesnik, 65, 4, 454-465 (2012) · Zbl 1299.30037 |
| [18] | Ismail, M. E. H.; Libis, C. A., Contiguous relations, basic hypergeometric functions, and orthogonal polynomials, I, Journal of Mathematical Analysis and Applications, 141, 2, 349-372 (1989) · Zbl 0681.33011 · doi:10.1016/0022-247x(89)90182-0 |
| [19] | Frank, E., A new class of continued fraction expansions for the ratios of Heine functions. II, Transactions of the American Mathematical Society, 95, 1, 17-26 (1960) · Zbl 0091.06403 · doi:10.1090/s0002-9947-1960-0116105-8 |
| [20] | Wall, H. S., Analytic Theory of Continued Fractions (1948), New York, NY, USA: D. Van Nostrand Company, New York, NY, USA · Zbl 0035.03601 |
| [21] | Liu, J.-G.; Pego, R. L., On generating functions of Hausdorff moment sequences, Transactions of the American Mathematical Society · Zbl 1350.44003 |
| [22] | Wall, H. S., Continued fractions and totally monotone sequences, Transactions of the American Mathematical Society, 48, 2, 165-184 (1940) · JFM 66.0255.01 · doi:10.1090/s0002-9947-1940-0002642-0 |
| [23] | Garabedian, H. L.; Wall, H. S., Hausdorff methods of summation and continued fractions, Transactions of the American Mathematical Society, 48, 185-207 (1940) · JFM 66.0255.02 · doi:10.1090/s0002-9947-1940-0002643-2 |
| [24] | Küstner, R., On the order of starlikeness of the shifted Gauss hypergeometric function, Journal of Mathematical Analysis and Applications, 334, 2, 1363-1385 (2007) · Zbl 1118.30010 · doi:10.1016/j.jmaa.2007.01.011 |
| [25] | Küstner, R., Hypergeometric functions and convolutions of starlike or convex functions of order \(α\), Computational Methods and Function Theory, 2, 2, 597-610 (2002) · Zbl 1053.30006 |
| [26] | Pólya, G., Application of theorem connected with the problem of moments, Messenger of Mathematics, 55, 189-192 (1925/1926) · JFM 52.0296.05 |
| [27] | Agrawal, S.; Sahoo, S. K., Geometric properties of basic hypergeometric functions, Journal of Difference Equations and Applications, 20, 11, 1502-1522 (2014) · Zbl 1304.30010 · doi:10.1080/10236198.2014.946501 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
