Let SH be the class of functions f = h+g that are harmonic univalent and sense-preserving in the open unit disk U = { z : |z| < 1} for which f (0) = f'(0)-1=0. In this paper, we introduce and study a subclass H( α, β) of the class SH and the subclass NH( α, β) with negative coefficients. We obtain basic results involving sufficient coefficient conditions for a function in the subclass H( α, β) and we show that these conditions are also necessary for negative coefficients, distortion bounds, extreme points, convolution and convex combinations. In this paper an attempt has also been made to discuss some results that uncover some of the connections of hypergeometric functions with a subclass of harmonic univalent functions.
@article{bwmeta1.element.doi-10_1515_math-2015-0066,
author = {Janusz Sok\'o\l\ and Rabha W. Ibrahim and M. Z. Ahmad and Hiba F. Al-Janaby},
title = {Inequalities of harmonic univalent functions with connections of hypergeometric functions},
journal = {Open Mathematics},
volume = {13},
year = {2015},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_math-2015-0066}
}
Janusz Sokół; Rabha W. Ibrahim; M. Z. Ahmad; Hiba F. Al-Janaby. Inequalities of harmonic univalent functions with connections of hypergeometric functions. Open Mathematics, Tome 13 (2015) . http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_math-2015-0066/
[1] Clunie J. and Sheil-Small T., Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A. I. Math., 1984, 9, 3–25 | Zbl 0506.30007
[2] Ahuja O. P. and Silverman H., Inequalities associating hypergeometric functions with planer harmonic mapping, J. Inequal Pure and Appl. Maths., 2004, 5, 1–10.
[3] Ahuja O. P., Planer harmonic convolution operators generated by hypergeometric functions, Trans. Spec. Funct., 2007, 18, 165–177. [Crossref] | Zbl 1114.30008
[4] Ahuja O. P., Connections between various subclasses of planer harmonic mappings involving hypergeometric functions, Appl. Math. Comput, 2008, 198, 305–316. [WoS] | Zbl 1138.31001
[5] Ahuja O. P., Harmonic starlikeness and convexity of integral operators generated by hypergeometric series, Integ. Trans. Spec. Funct., 2009, 20(8), 629–641. [Crossref][WoS] | Zbl 1203.30012
[6] Ahuja O. P., Inclusion theorems involving uniformly harmonic starlike mappings and hypergeometric functions, Analele Universitatii Oradea, Fasc. Matematica, Tom XVIII, 2011, 5–18. | Zbl 1249.30017
[7] Murugusundaramoorthy G. and Raina R. K., On a subclass of harmonic functions associated with the Wright’s generalized hypergeometric functions, Hacet. J. Math. Stat., 2009, 28(2), 129–136. | Zbl 1178.30014
[8] Sharma P., Some Wgh inequalities for univalent harmonic analytic functions, Appl. Math, 2010, 464–469. [Crossref]
[9] Raina R. K. and Sharma P., Harmonic univalent functions associated with Wright’s generalized hypergeometric functions, Integ. Trans. Spec. Funct., 2011, 22(8), 561–572. [Crossref][WoS] | Zbl 1243.30038
[10] Ahuja O. P. and Sharma P., Inclusion theorems involving Wright’s generalized hypergeometric functions and harmonic univalent functions, Acta Univ. Apulensis, 2012, 32, 111–128. | Zbl 1313.30022
[11] Avci Y. and Zlotkiewicz E., On harmonic univalent mappings, Ann. Univ. Marie Curie-Sklodowska Sect. A, 1991, 44, 1–7. | Zbl 0780.30013
[12] Silverman H., Harmonic univalent functions with negative coefficients, J. Math. Anal. Apl., 1998, 220, 283–289. | Zbl 0908.30013
[13] Silverman H. and Silvia E. M., Subclasses of harmonic univalent functions, New Zealand J. Math., 1999, 28, 275–284. | Zbl 0959.30003
[14] Ahuja O. P. and Jahangiri J. M., Noshiro-Type harmonic univalent functions, Sci. Math. Jpn., 2002, 6(2), 253–259. | Zbl 1018.30006
[15] Yalcin S.,Ozturk M. and Yamankaradeniz M., A subclass of harmonic univalent functions with negative coefficients, Appl. Math. Comput., 2003, 142, 469-–476. [WoS] | Zbl 1033.30011
[16] AL-Khal R. A. and AL-Kharsani H.A., Harmonic hypergeometric functions, Tamkang J. Math., 2006, 37(3), 273–283. | Zbl 1123.30009
[17] Yalcin S. and Ozturk M., A new subclass of complex harmonic functions, Math. Inequal. Appl., 2004, 7(1), 55–61. | Zbl 1056.30014
[18] Chandrashekar R., Lee S. K. and Subramanian K. G., Hyergeometric functions and subclasses of harmonic mappings, Proceeding of the International Conference on mathematical Analysis 2010, Bangkok, 2010, 95–103.
[19] Dixit R. K.K., Pathak A. L., Powal S. and Agarwal R., On a subclass of harmonic univalent functions defined by convolution and integral convolution, I. J. Pure Appl. Math., 2011, 69(3), 255–264. | Zbl 1220.30019
[20] Aouf M.K., Mostafa A. O., Shamandy A. and Adwan E. A., Subclass of harmonic univalent functions defined by Dziok-Srivastava operators, Le Math., 2013, LXVIII, 165–177. | Zbl 1296.30012
[21] El-Ashwah R.M., Subclass of univalent harmonic functions defined by dual convolution, J. Inequal. Appl., 2013, 2013:537, 1–10. [Crossref][WoS] | Zbl 1293.31002
[22] Al-Khal R. A. and AL-Kharsani H. A., Harmonic univalent functions, J. Inequal. Pure Appl. Math., 2007, 8(2), 1–8. | Zbl 1139.30004
[23] Nagpal S. and Ravichandran V., Univalence and convexity in one direction of the convolution of harmonic mappings, Complex Var. Elliptic Equ., 2014, 59(7), 1328–1341. [WoS]
[24] Ponnusamy S.,Rasila A. and Kaliraj A., Harmonic close-to-convex functions and minimal surfaces, Complex Var. Elliptic Equ., 2014, 59(9), 986–1002. | Zbl 1297.31003
[25] Porwal S. and Dixit K. K., An application of hypergeometric functions on harmonic univalent functions, Bull. Math. Anal. Appl., 2010, 2(4), 97–105. | Zbl 1312.30030
[26] Shelake G., Joshi S., Halim S., On a subclass of harmonic univalent functions defined by convolution, Acta Univ. Apulensis, 2014, 38, 251–262. | Zbl 1340.30076