Inequalities of harmonic univalent functions with connections of hypergeometric functions

Janusz Sokół; Rabha W. Ibrahim; M. Z. Ahmad; Hiba F. Al-Janaby

Open Mathematics (2015)

  • Volume: 13, Issue: 1
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Janusz Sokół, et al. "Inequalities of harmonic univalent functions with connections of hypergeometric functions." Open Mathematics 13.1 (2015): null. <http://eudml.org/doc/275864>.

@article{JanuszSokół2015,
abstract = {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.},
author = {Janusz Sokół, Rabha W. Ibrahim, M. Z. Ahmad, Hiba F. Al-Janaby},
journal = {Open Mathematics},
keywords = {Harmonic function; Analytic function; Univalent function; Unit disk},
language = {eng},
number = {1},
pages = {null},
title = {Inequalities of harmonic univalent functions with connections of hypergeometric functions},
url = {http://eudml.org/doc/275864},
volume = {13},
year = {2015},
}

TY - JOUR
AU - Janusz Sokół
AU - Rabha W. Ibrahim
AU - M. Z. Ahmad
AU - Hiba F. Al-Janaby
TI - Inequalities of harmonic univalent functions with connections of hypergeometric functions
JO - Open Mathematics
PY - 2015
VL - 13
IS - 1
SP - null
AB - 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.
LA - eng
KW - Harmonic function; Analytic function; Univalent function; Unit disk
UR - http://eudml.org/doc/275864
ER -

References

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  1. [1] Clunie J. and Sheil-Small T., Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A. I. Math., 1984, 9, 3–25 Zbl0506.30007
  2. [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. [3] Ahuja O. P., Planer harmonic convolution operators generated by hypergeometric functions, Trans. Spec. Funct., 2007, 18, 165–177. [Crossref] Zbl1114.30008
  4. [4] Ahuja O. P., Connections between various subclasses of planer harmonic mappings involving hypergeometric functions, Appl. Math. Comput, 2008, 198, 305–316. [WoS] Zbl1138.31001
  5. [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] Zbl1203.30012
  6. [6] Ahuja O. P., Inclusion theorems involving uniformly harmonic starlike mappings and hypergeometric functions, Analele Universitatii Oradea, Fasc. Matematica, Tom XVIII, 2011, 5–18. Zbl1249.30017
  7. [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. Zbl1178.30014
  8. [8] Sharma P., Some Wgh inequalities for univalent harmonic analytic functions, Appl. Math, 2010, 464–469. [Crossref] 
  9. [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] Zbl1243.30038
  10. [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. Zbl1313.30022
  11. [11] Avci Y. and Zlotkiewicz E., On harmonic univalent mappings, Ann. Univ. Marie Curie-Sklodowska Sect. A, 1991, 44, 1–7. Zbl0780.30013
  12. [12] Silverman H., Harmonic univalent functions with negative coefficients, J. Math. Anal. Apl., 1998, 220, 283–289. Zbl0908.30013
  13. [13] Silverman H. and Silvia E. M., Subclasses of harmonic univalent functions, New Zealand J. Math., 1999, 28, 275–284. Zbl0959.30003
  14. [14] Ahuja O. P. and Jahangiri J. M., Noshiro-Type harmonic univalent functions, Sci. Math. Jpn., 2002, 6(2), 253–259. Zbl1018.30006
  15. [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] Zbl1033.30011
  16. [16] AL-Khal R. A. and AL-Kharsani H.A., Harmonic hypergeometric functions, Tamkang J. Math., 2006, 37(3), 273–283. Zbl1123.30009
  17. [17] Yalcin S. and Ozturk M., A new subclass of complex harmonic functions, Math. Inequal. Appl., 2004, 7(1), 55–61. Zbl1056.30014
  18. [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. [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. Zbl1220.30019
  20. [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. Zbl1296.30012
  21. [21] El-Ashwah R.M., Subclass of univalent harmonic functions defined by dual convolution, J. Inequal. Appl., 2013, 2013:537, 1–10. [Crossref][WoS] Zbl1293.31002
  22. [22] Al-Khal R. A. and AL-Kharsani H. A., Harmonic univalent functions, J. Inequal. Pure Appl. Math., 2007, 8(2), 1–8. Zbl1139.30004
  23. [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. [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. Zbl1297.31003
  25. [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. Zbl1312.30030
  26. [26] Shelake G., Joshi S., Halim S., On a subclass of harmonic univalent functions defined by convolution, Acta Univ. Apulensis, 2014, 38, 251–262. Zbl1340.30076

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